Shape the ground chuck into a patty, making sure it is tightly packed together. Season both sides of the burger generously. Take a small glass and cut out a hole in the middle of the burger.
Break the egg into a ramekin.
Preheat a heavy cast iron skillet over medium high heat and add the burger patty with the hole cut out. Cook until browned and crisp on one side, 1-3 minutes.
Sprinkle 2 tablespoons of shredded cheese in the rough shape of your burger bun into a non-stick pan. Add the bun on top and grill on medium-low until the cheese is crispy and golden.
Check the burger using a spatula. When brown, carefully flip the burger. Pour the egg into the hole. If it looks like the egg white will overflow, stop pouring and throw away remaining egg white. Cook burger on medium-low, covered, until the egg is done to your liking.
Sprinkle 2 tablespoons of shredded cheese in the rough shape of the remaining half of the burger bun into the non-stick pan. Add the remaining bun half and grill on medium-low until the cheese is crispy and golden.
When the burger and egg are done, season with salt and pepper, and place on top of grilled cheese buns. Top with desired toppings and serve immediately.