Shape the beef into 4 patties, each about 5-inches in diameter. Sprinkle the outside of the patties generously with salt and pepper and set aside.
Lay the bacon in a large nonstick skillet and cook over medium heat, until brown and crispy, about 4 minutes per side. Transfer to a paper-towel-lined plate to drain. Pour off and reserve all but about 1 tablespoon of the bacon drippings and increase the heat to medium-high.
Add the burgers to the skillet, and cook until well browned on the outside and medium-rare on the inside, about 4 minutes per side. Transfer the cooked burgers to a rimmed baking sheet and top each with a slice of cheese. Brush the insides of the buns with the melted butter and put on a second baking sheet buttered side-up. Bake until the cheese melts, about 3 minutes and the buns are lightly toasted, about 10 minutes.
Meanwhile, wipe out the skillet. Heat 1 tablespoon of the reserved bacon drippings over medium-high heat. Break the eggs into the skillet and cook until the whites are set and the yolks are still runny, about 5 minutes. Season with salt and pepper.
Put a burger patty on each of the bottom buns and top each with 2 slices of bacon and a fried egg. Spread the top bun with some of the spicy ketchup and put on top of the burger.
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