How do I keep the burger from falling apart?

I would guess that the beans were too dry. How long did you soak them? Some beans are dryer than others when you bring them home from the store, so perhaps they needed a longer soaking time.

Do you need the recipe to be vegan? Adding an egg would help hold things together.

Finally, if I were making this recipe, I would not use uncooked soaked beans. To avoid mushiness, I'd do what Kenji Lopez-Alt does in his bean burger recipe: roast canned beans in the oven just long enough to dry them out a bit (he recommends in the "Really Awesome Black Beans Burger" recipe to roast at 350 degrees for 20 minutes - hands off time that's well worth the effort). It works like a charm!

;o)