Grind it!
Do it fast, so the grinder doesn't heat up enough (KitchenAid attachment manual says speed 4; I do 6). Helps to put grind into bowl over ice bath to keep it cold. Measure out 6 to 6 1/2 oz patties, or, if it's the weekend and you hate yourself, bigger. Just don't smash them too much.
Also I'd do without the egg wash on the bun. Anyway, ask 10,000 chefs how to do something and you'll get 10,000 different answers.