Heat a gas grill to high heat. Grill the first side for 3 minutes or until golden brown and slightly charred. Flip over burgers. Cook 6 minutes or until it reaches desired doneness. When you have one minute left of grilling time place a slice of cheese on top of burger and let it melt.
While burgers are cooking, fry your eggs either on a skillet on the stove or on the grill. Heat a pan with 2 Tbsp butter over medium high heat. When the butter foams, break the eggs into the pan, spacing them about 2 inches apart.
Reduce the heat to low and season the eggs with salt and pepper. Cover the pan and cook until the whites are set and the yolks are just beginning to firm around the edge, about 5 minutes.
While burgers are cooking butter the insides of the buns with 2 Tbsp butter. Place buns, butter side down, on grill grates to toast. Remove from grill when they are lightly toasted.
When making your own burgers, use beef that is no leaner than 85%. Due to the high fat content, your burger will be juicy and full of flavor. Beware that higher fat meat will shrink more when they are grilled.