Use a gentle hand when forming the patty; you want it to be tender and juicy, not dense. After stacking the burger patty on the bun with the cheese, return to the oven for 1 to 2 minutes to get a gooey melt, and then resume building your burger.
Preheat the oven to 350 degrees F. Mix the ketchup, brown sugar, Worcestershire sauce, chili powder and hot sauce in a small bowl. Set aside.
Using a fork, gently mix the beef with the paprika, 1 teaspoon salt and 1/2 teaspoon black pepper. Form into four 5-inch patties. Cover and set aside.
Cook the bacon in a large nonstick skillet over medium heat until the fat is rendered and the bacon is crisp, about 8 minutes. Remove and drain on a paper-towel-lined plate. Pour off most of the bacon fat, leaving about 2 teaspoons in the skillet, reserving the extra bacon fat. Heat the skillet over medium-high heat. Add the burgers to the skillet, 2 at a time, and cook until nicely browned, 4 minutes. Flip and cook the other side until the burger is done, about 4 minutes for medium-rare. Repeat with the remaining burgers. Place the burgers on a rimmed baking sheet and place in the oven to keep warm while cooking the eggs.
Wipe out the skillet. Add 1 tablespoon of the reserved bacon fat to the skillet and heat over medium-high heat. Break the eggs into the skillet and cook until the whites are set, about 5 minutes. Season with salt and pepper.
Meanwhile, brush the buns with melted butter. Place them on a rimmed baking sheet and bake until lightly toasted, about 10 minutes.
Build the burger by placing a hamburger patty on the bottom bun, add a slice of cheese, 2 pieces of bacon and top with a fried egg. Spread the top bun with some of the spiced ketchup and place on top.