VNC-Server as a virtual X11 monitor to expand screen

tl;dr: Force a "virtual" output of your gfx card to a display mode, and export that with x11vnc.

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You can achieve this, but there are a few prerequisites:

• A graphics card with multi-head capabilities (= can render several "desktop" surfaces). Which is most cards these days.
• x11vnc, a mature software (x11vnc) to export X11 surfaces (among others) to VNC clients.

Most consumer cards these days can render several different outputs. Mine can do 3, out of the 5 that xrandr shows (eDP1,HDMI[12],DP[12]).

• Pick an unused output from xrandr, in my example HDMI2.

• Pick a resolution for the screen of the vnc client, and generate a mode:

  $ cvt 1920 1080
# 1920x1080 59.96 Hz (CVT 2.07M9) hsync: 67.16 kHz; pclk: 173.00 MHz
Modeline "1920x1080_60.00" 173.00 1920 2048 2248 2576 1080 1083 1088 1120 -hsync +vsync

• Add that mode to xrandr

  xrandr --newmode "1920x1080_60" 173.00 1920 2048 2248 2576 1080 1083 1088 1120 -hsync +vsync

• Put e.g. HDMI2 in that mode, and attach to the right of eDP1 (Main screen)

  xrandr --addmode HDMI2 1920x1080_60
--output HDMI2 --mode 1920x1080_60 --right-of eDP1

• Now export that with x11vnc, choosing the appropriate offset:

  x11vnc -display :0 -clip 1920x1080+1600+0 <other options>

  Note: Add desired encryption/authentication/other options to that command.

• Now connect to with a VNC client to your "virtual monitor". (or modify above command to connect to a "listening" VNC-client.

See this answer to be able to use VIRTUAL1 instead of a real output like HDMI2, and be able to manage it (select resolution, arrange desktop geometry, clone or extend) in the desktop environment screen settings like any other physical monitor.