Is it possible to create extension constructors in Kotlin?

Seems that there are not an official "function extension for constructors" but you can create a package-method that imitates a constructor

class Foo() {
    ...
}

fun Foo(stuff: Int): Foo = Foo().apply {setStuff(stuff)}

fun main(args: Array<String>){
    println(Foo(123))
}

Based on @s1m0nw1 solution, I can adjust it to be more like the OP wanted by overriding operator invoke.

// base class
class Whatever() {
    companion object {
        fun setPotato(potato: String)
    }
}

operator fun Whatever.Companion.invoke(potato: String) {
    setPotato(potato)
}

fun main(args: Array<String>) {
    println(Whatever("holi"))
}

Keep in mind that it still needs companion object inside base class. If you can't edit the source of base class (in case of kotlin framework class or third party class), and if it already has companion object that you want to add your new method to, you can always use an extension function:

fun Int.Companion.setPotato(potato: String) {
    // ...
}

Not like in Swift, because:

Extensions are resolved statically. Extensions do not actually modify classes they extend. By defining an extension, you do not insert new members into a class, but merely make new functions callable with the dot-notation on variables of this type. (Source)

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If a companion object is defined in your target class, go with s1m0nw1's approach. The advantage is that you can call the extension function without an instance (statically) of the target class.

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If not, use a classic Factory Pattern:

class Fruit(var name: String = "") {
}

class FruitFactory {
    companion object {
        fun create(name: String): Fruit {
            return Fruit().apply {
                this.name = "Tasty $name"
            }
         }
    }
}

fun main(args: Array<String>) {
    val orange = Fruit("Orange")
    println(orange.name)

    val apple = FruitFactory.create("Apple")
    println(apple.name)
}

You can extend the Factory as you wish with further constructors either nested or as extension functions.

Output:

Orange
Tasty Apple

You can't do this. What you can do though: Extend the companion object of a class with a factory method:

// base class
class Whatever() {
    companion object {

    }

}

// factory extension
fun Whatever.Companion.withPotato(potato: String) {
    //setPotato(potato)
}

fun main(args: Array<String>) {
    println(Whatever.withPotato("holi"))
}

The only problem: A companion object has to be existent to do this.