You should add egg to hamburger if you want the meat to bind and keep the burgers from falling apart and if you're also adding other ingredients to the ground beef like diced onions or breadcrumbs.
However most people prefer to skip the eggs when making hamburgers as it results in a simpler burger.
And some people prefer the hamburgers without eggs and find the hamburgers with eggs to be an inferior burger, but others like it.
The protein in the eggs when added to the hamburgers helps with the structure of the meat and makes them great for crumbly burgers.
Although if you handle the hamburger meat minimally and form a good crust when cooking then you can leave the eggs out.
The eggs added to the hamburgers, basically help hold the hamburger patty together, especially when adding onions, peppers, breadcrumbs etc that add moisture and also separate the fat.
And the egg yolk also can add richness and flavor and turn the hamburger into a miniature meatloaf texture.
But if you want a pure beef flavor and texture, it's best to leave out the eggs and just use some salt and pepper and avoid overmixing the meat and handle it gently to form the hamburger patties so they won't crumble and don't flip the hamburgers too often.
If you add eggs to the hamburgers you should use 1 egg per lb of meat, or just an egg yolk and let the formed hamburger patties rest in the refrigerator for 15 minutes to 30 minutes before you cook them to help the hamburger patties firm up.
When grilling burgers or cooking burgers you can also use the 5 6 7 rule for burgers which is a guideline for grilling burgers, which helps you achieve consistent doneness of the grilled burger, which is based on cooking time per side.
This is assuming you have a 1/2 inch to 3/4 inch thick hamburger patty, over medium heat.
The 5 6 7 rule for burgers suggests that you grill the burgers for 5 minutes per side for rare to medium rare, 6 minutes for medium and grill the burgers for 7 minutes for well done.
This helps to ensure that you get a nice juicy burger on the grill without needing a thermometer.
5 minutes of cooking per side when grilling the burgers, produces a burger that is rare to medium rare.
6 minutes of cooking per side produces a burger that is medium done.
And 7 minutes of cooking per side produces a burger on the grill that is well done.
So if you want your grilled burger to be rare to medium rare, you would grill the burgers for 5 minutes per side.
If you want your grilled burger to be medium done, you would grill your burger for 6 minutes per side.
And if you want your grilled burger to be done fully, then you would grill your burger for 7 minutes per side.
Also don't press down on the burgers when grilling them, to prevent drying the burgers out.
And the 5 6 7 rule for grilling burgers also works best on a grill that is preheated to 350 F to 400 F with 80/20 ground beef.
And some interpretations of the 5 6 7 rule, define it as 5 minutes on one side, 6 minutes on the other side and a 7 minute rest period for maximum juiciness.