Divide and shape ground beef into three patties and season both sides with salt and pepper. On a grill or pan, cook your burgers to desired temperature. After flipping, place thinly sliced cheese on each burger for melting. Toast the buns. While the burgers rest, you can fry the eggs.
Place a cast iron pan directly on the grill or stove over medium high heat. Melt one tablespoon of butter and crack 1 egg into the hot pan. Lightly fry until the whites have set, about two minutes. Tilt the pan to the side and spoon melted butter over top of the yolk until it reaches your desired runniness. Finish with a pinch of sea salt. Repeat with each additional egg.
Spread toasted buns with your favorite burger condiments and assemble each burger with a few pieces of lettuce and a fried egg.