Make two hamburger patties by forming ⅓ pound of ground beef into a ball, then press flat. Season each side liberally with salt and black pepper.
Set a large skillet on the stovetop over medium high heat. Place the patties in the skillet and cook until the bottom forms a nice sear, approximately three minutes. Flip the patties and top with a cheddar cheese slice. Cook for three additional minutes, or until the interior reaches your desired doneness.
While the patties cook, spread two tablespoons of butter across the interior of the hamburger buns, then toast in a separate skillet set over medium heat.
Remove the buns from the skillet and add an additional tablespoon of butter. Crack the eggs into the skillet, sprinkle with salt and black pepper, and cook until the bottom of the egg has set. Flip the eggs, and cook for an additional minute, or until the yolk reaches your desired doneness.
Assemble the burgers by placing a lettuce leaf on the bottom bun, top with thin slices of onion, the burger patty, cheese, fried egg, and the top bun.
Let the burger rest for a minute or two, then serve.