test.ts

interface IMenu {
  stuff?: any;
  items: string[];
}

interface IButton {
...
  menuProps?: IMenu
}

interface IMenuBar {
  overflowButtonProps?: IButton // but menuProps should be Partial<IMenu> because IMenuBar provides required items
}

This isn't super elegant, but it seems to do the trick for the stated use case.

interface IMenu {
  stuff?: any;
  items: string[];
}

interface IButton<IMenuType extends Partial<IMenu> = IMenu> {
  menuProps?: IMenuType
}

interface IMenuBar {
  overflowButtonProps?: IButton<Partial<IMenu>>
}