Cook Burger on Stove - Turns Gray
If you add egg and breadcrumbs, you aren't cooking a burger, you're cooking meatloaf. 

Try using just good beef with plenty of fat (80% meat, 20% fat is the ideal ratio, lean mince is not good for burgers), and pepper. Get the pan good and hot, salt the burgers well (don't add salt beforehand), brush the burgers with oil, then whack them on the pan, 5-6 minutes a side (depends on thickness) and you'll get a nice brown crust.

A couple of other tips: make the burgers plenty ahead of time so they can sit in the fridge for a good few hours. And never press the burger down with your spatula, you just end up losing all the juiciness. 

Check out The Burger Lab's Top Ten Tips for a Better Burger for more tips.