dropdown list in editor of functions from linked script

I'm after the same kind of functionality that is used on UI buttons, where you can drag a target object into linked area, then choose from the functions contained on that object's script.

Unity's version looks like this:

I am planning on using this functionality on a generic menu, so buttons are menu options, access with arrow keys and action button. The menus can be any length, so the options need to populate based on the array length.

Something like this, as a resulting view of the editor (mockup):

So you would basically drag in a target (Game) and it would list out the functions from any scripts attached via a dropdown, then when that button is active and the user presses a button on the joypad it runs that function.

I'm not sure where to really start with this kind of setup, as I don't know what this functionality is called in technical terms.

Using C#