Meat
Grind the best quality chuck roast you can find, and you're set. Actually, you don't even need a grinder (even though they're cheap); you can chop/mince cold beef and get arguably better texture. It just takes longer. I like a mix of about 60 percent chuck and 40 percent brisket.
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.