Open second drawerlayout as a sub drawerlayout over the first

It's unclear how exactly you wish to implement your drawer UI, so the following solution is rather generic, in that it should work with NavigationViews, RecyclerViews, or pretty much whatever type of Views you'd like.

This solution uses a custom ViewSwitcher subclass that acts as a DrawerLayout's left drawer, and holds two child Views, one being the main drawer View, and the other being the second drawer that opens over it.

The DoubleDrawerView class is a relatively simple ViewSwitcher that loads its own Animations, and juggles them appropriately to give the effect of a second drawer opening and closing over the first. It tracks its own state so that it can be restored correctly after a device rotation, etc.

public class DoubleDrawerView extends ViewSwitcher {
    private static final int NONE = -1;
    private static final int MAIN_VIEW_INDEX = 0;
    private static final int DRAWER_VIEW_INDEX = 1;

    private Animation slideInAnimation, slideOutAnimation, noAnimation;
    private boolean animating = false;

    private Animation.AnimationListener listener = new Animation.AnimationListener() {
        @Override
        public void onAnimationEnd(Animation anim) {
            animating = false;
        }

        @Override
        public void onAnimationStart(Animation anim) {}

        @Override
        public void onAnimationRepeat(Animation anim) {}
    };

    public DoubleDrawerView(Context context) {
        this(context, null);
    }

    public DoubleDrawerView(Context context, AttributeSet attrs) {
        super(context, attrs);

        slideInAnimation = AnimationUtils.loadAnimation(context, R.anim.slide_in_left);
        slideOutAnimation = AnimationUtils.loadAnimation(context, R.anim.slide_out_left);
        noAnimation = AnimationUtils.loadAnimation(context, R.anim.none);
        noAnimation.setAnimationListener(listener);
    }

    public void openInnerDrawer() {
        if (getDisplayedChild() != DRAWER_VIEW_INDEX) {
            setChildAndAnimate(DRAWER_VIEW_INDEX, true);
        }
    }

    public void closeInnerDrawer() {
        if (getDisplayedChild() != MAIN_VIEW_INDEX) {
            setChildAndAnimate(MAIN_VIEW_INDEX, true);
        }
    }

    public boolean isInnerDrawerOpen() {
        return getDisplayedChild() == DRAWER_VIEW_INDEX;
    }

    private void setChildAndAnimate(int whichChild, boolean doAnimate) {
        if (doAnimate) {
            setAnimationForChild(whichChild);
        }
        else {
            setAnimationForChild(NONE);
        }
        animating = doAnimate;
        setDisplayedChild(whichChild);
    }

    private void setAnimationForChild(int whichChild) {
        if (whichChild == DRAWER_VIEW_INDEX) {
            setInAnimation(slideInAnimation);
            setOutAnimation(noAnimation);
        }
        else if (whichChild == MAIN_VIEW_INDEX) {
            setInAnimation(noAnimation);
            setOutAnimation(slideOutAnimation);
        }
        else {
            setInAnimation(null);
            setOutAnimation(null);
        }
    }

    @Override
    public boolean onInterceptTouchEvent(MotionEvent ev) {
        if (animating) {
            return true;
        }
        else {
            return super.onInterceptTouchEvent(ev);
        }
    }

    @Override
    protected Parcelable onSaveInstanceState() {
        Parcelable superState = super.onSaveInstanceState();
        SavedState ss = new SavedState(superState);
        ss.whichChild = getDisplayedChild();
        return ss;
    }

    @Override
    protected void onRestoreInstanceState(Parcelable state) {
        SavedState ss = (SavedState) state;
        super.onRestoreInstanceState(ss.getSuperState());
        setChildAndAnimate(ss.whichChild, false);
    }

    private static class SavedState extends BaseSavedState {
        int whichChild;

        SavedState(Parcelable superState) {
            super(superState);
        }

        private SavedState(Parcel in) {
            super(in);
            whichChild = in.readInt();
        }

        @Override
        public void writeToParcel(Parcel out, int flags) {
            super.writeToParcel(out, flags);
            out.writeInt(whichChild);
        }

        public static final Parcelable.Creator<SavedState>
            CREATOR = new Parcelable.Creator<SavedState>() {

            public SavedState createFromParcel(Parcel in) {
                return new SavedState(in);
            }

            public SavedState[] newArray(int size) {
                return new SavedState[size];
            }
        };
    }
}

DoubleDrawerView uses the following XML files for its Animations. These should be in your project's res/anim/ folder.

slide_in_left.xml

<translate xmlns:android="http://schemas.android.com/apk/res/android"
    android:fromXDelta="-100%p" android:toXDelta="0"
    android:duration="@android:integer/config_mediumAnimTime"/>

slide_out_left.xml

<translate xmlns:android="http://schemas.android.com/apk/res/android"
    android:fromXDelta="0" android:toXDelta="-100%p"
    android:duration="@android:integer/config_mediumAnimTime"/>

none.xml

<alpha xmlns:android="http://schemas.android.com/apk/res/android"
    android:fromAlpha="1.0" android:toAlpha="1.0"
    android:duration="@android:integer/config_mediumAnimTime" />

This example's layout is a standard DrawerLayout with a DoubleDrawerView for its drawer, and two simple NavigationViews therein. Do note that the main drawer View must be listed first inside the DoubleDrawerView, with the second, inner drawer View after.

activity_main.xml

<android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:id="@+id/drawer_layout"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

    <FrameLayout
        android:id="@+id/main_content"
        android:layout_width="match_parent"
        android:layout_height="match_parent" />

    <com.example.doubledrawer.DoubleDrawerView
        android:id="@+id/double_drawer_view"
        android:layout_width="240dp"
        android:layout_height="match_parent"
        android:layout_gravity="left">

        <android.support.design.widget.NavigationView
            android:id="@+id/main_navigation_view"
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            app:menu="@menu/navigation_main" />

        <android.support.design.widget.NavigationView
            android:id="@+id/settings_navigation_view"
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            app:menu="@menu/navigation_settings" />

    </com.example.doubledrawer.DoubleDrawerView>

</android.support.v4.widget.DrawerLayout>

For the sake of a complete cut and paste example, some simple res/menu/ files for the NavigationViews above.

navigation_main.xml

<menu xmlns:android="http://schemas.android.com/apk/res/android">

    <group
        android:id="@+id/group_screens"
        android:checkableBehavior="single">
        <item
            android:id="@+id/menu_screen_1"
            android:title="Screen 1" />
        <item
            android:id="@+id/menu_screen_2"
            android:title="Screen 2"/>
    </group>

    <item
        android:id="@+id/menu_open_settings"
        android:title="Open Settings" />

</menu>

navigation_settings.xml

<menu xmlns:android="http://schemas.android.com/apk/res/android">

    <item
        android:id="@+id/menu_close_settings"
        android:title="Back to Main" />

    <group
        android:id="@+id/group_settings">
        <item
            android:id="@+id/menu_setting_1"
            android:title="Setting 1" />
        <item
            android:id="@+id/menu_setting_2"
            android:title="Setting 2" />
    </group>

</menu>

In the example Activity, we just get references to the DoubleDrawerView and NavigationViews, and implement an OnNavigationItemSelectedListener to open and close the inner drawer accordingly.

public class MainActivity extends AppCompatActivity
    implements NavigationView.OnNavigationItemSelectedListener {

    private DrawerLayout drawerLayout;
    private DoubleDrawerView doubleDrawerView;
    private NavigationView mainNavigationView, settingsNavigationView;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        drawerLayout = (DrawerLayout) findViewById(R.id.drawer_layout);
        doubleDrawerView = (DoubleDrawerView) findViewById(R.id.double_drawer_view);
        mainNavigationView = (NavigationView) findViewById(R.id.main_navigation_view);
        settingsNavigationView = (NavigationView) findViewById(R.id.settings_navigation_view);

        mainNavigationView.setNavigationItemSelectedListener(this);
        settingsNavigationView.setNavigationItemSelectedListener(this);

        drawerLayout.openDrawer(Gravity.LEFT);
    }

    @Override
    public boolean onNavigationItemSelected(MenuItem item) {
        switch (item.getItemId()) {
            case R.id.menu_open_settings:
                doubleDrawerView.openInnerDrawer();
                break;

            case R.id.menu_close_settings:
                doubleDrawerView.closeInnerDrawer();
                break;

                // Additional cases as needed
                // This example simply Toasts the title for the extra sample items

            default:
                Toast.makeText(this, item.getTitle(), Toast.LENGTH_SHORT).show();
        }
        return true;
    }
}

This solution uses two DrawerLayouts, one nested in the other, in lieu of a custom View. This might be slightly easier to implement, but it requires a bit more specialized code in the Activity itself, and therefore will be more tightly coupled to whichever classes it's used in.

The Activity's layout, activity_main.xml.

<android.support.v4.widget.DrawerLayout
    xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:app="http://schemas.android.com/apk/res-auto"
    android:id="@+id/drawer_layout"
    android:layout_width="match_parent"
    android:layout_height="match_parent">

    <FrameLayout
        android:id="@+id/main_content"
        android:layout_width="match_parent"
        android:layout_height="match_parent" />

    <android.support.v4.widget.DrawerLayout
        android:id="@+id/inner_drawer_layout"
        android:layout_width="240dp"
        android:layout_height="match_parent"
        android:layout_gravity="left">

        <android.support.design.widget.NavigationView
            android:id="@+id/main_navigation_view"
            android:layout_width="match_parent"
            android:layout_height="match_parent"
            app:menu="@menu/navigation_main" />

        <android.support.design.widget.NavigationView
            android:id="@+id/settings_navigation_view"
            android:layout_width="240dp"
            android:layout_height="match_parent"
            android:layout_gravity="left"
            app:menu="@menu/navigation_settings" />

    </android.support.v4.widget.DrawerLayout>

</android.support.v4.widget.DrawerLayout>

The example NavigationViews above use the same menu files as shown in my other answer here.

In the Activity, we get references to both DrawerLayouts, and set the scrim color and lock mode on the inner one at startup. We also need to handle the back button press ourselves, as the addition of a second DrawerLayout messes with the first's handling of it. Upon opening and closing the inner drawer, we need to set the lock mode appropriately, to prevent dragging the inner drawer.

public class MainActivity extends AppCompatActivity
    implements NavigationView.OnNavigationItemSelectedListener {

    private DrawerLayout drawerLayout, innerDrawerLayout;
    private NavigationView mainNavigationView, settingsNavigationView;

    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);

        drawerLayout = (DrawerLayout) findViewById(R.id.drawer_layout);
        innerDrawerLayout = (DrawerLayout) findViewById(R.id.inner_drawer_layout);
        mainNavigationView = (NavigationView) findViewById(R.id.main_navigation_view);
        settingsNavigationView = (NavigationView) findViewById(R.id.settings_navigation_view);

        mainNavigationView.setNavigationItemSelectedListener(this);
        settingsNavigationView.setNavigationItemSelectedListener(this);

        innerDrawerLayout.setScrimColor(Color.TRANSPARENT);
        innerDrawerLayout.setDrawerLockMode(DrawerLayout.LOCK_MODE_LOCKED_CLOSED);

        drawerLayout.openDrawer(Gravity.LEFT);
    }

    @Override
    public void onBackPressed() {
        if (drawerLayout.isDrawerOpen(Gravity.LEFT)) {
            drawerLayout.closeDrawer(Gravity.LEFT);
        }
        else {
            super.onBackPressed();
        }
    }

    private void openInnerDrawer() {
        innerDrawerLayout.openDrawer(Gravity.LEFT);
        innerDrawerLayout.setDrawerLockMode(DrawerLayout.LOCK_MODE_LOCKED_OPEN);
    }

    private void closeInnerDrawer() {
        innerDrawerLayout.closeDrawer(Gravity.LEFT);
        innerDrawerLayout.setDrawerLockMode(DrawerLayout.LOCK_MODE_LOCKED_CLOSED);
    }

    @Override
    public boolean onNavigationItemSelected(MenuItem item) {
        switch (item.getItemId()) {
            case R.id.menu_open_settings:
                openInnerDrawer();
                break;

            case R.id.menu_close_settings:
                closeInnerDrawer();
                break;

            // Additional cases as needed
            // This example simply Toasts the title for the extra sample items

            default:
                Toast.makeText(this, item.getTitle(), Toast.LENGTH_SHORT).show();
        }
        return true;
    }
}