Personally would add a bit of breadcrumbs, 2 eggs is more than enough. Then season. Seasoning is where you can play around. I've often added worcestershire or soy or both and just added one egg
One egg white mate, with 520 grams you will be able to make 3 quarter pounders Yes, but pointless if your pan isn’t smoking hot as the steak will not sear and will look and taste like a grey piece of cardboard.
That will ensure the moisture is kept inside. All depends on the cut of the beef mince you use, the egg white is the actual binding agent but cuts that include a higher fat content don’t need it but hey if you’ve cracked the egg why not use it all.
Generally, as long as you don't handle them too much during the shaping stage and chill them before cooking you shouldn't need an egg.