The Firefox add-on SDK allows for simple action buttons and toggle buttons. This creates a menu-button instead, which displays an icon on the left and a drop-marker on the right, like so:

Place both files in the lib directory of your extension.
Usage is similar to other button types. Include the following code in main.js (or any js file in the lib directory)

const { MenuButton } = require('./menu-button');
var btn = MenuButton({
  id: 'my-menu-button',
  label: 'My menu-button',
  icon: {
    "16": "./firefox-16.png",
    "32": "./firefox-32.png"
  },
  onClick: click
});

The click function will be passed the same state object as the toggle and action buttons, and will be passed an additional boolean argument: isMenu. It should be used like so

function click(state, isMenu) {
  if (isMenu) {
    //menu-button clicked
  } else {
    //icon clicked
  }
}