Cook the bacon in batches in a large skillet over medium heat until crisp, about 3 minutes per side. Remove to a paper towel-lined plate. Do not wipe out the skillet.
Shape the ground beef with your hands into 4 round patties about 1 1/2 inches thick and season each burger on both sides with the salt and pepper. Make a divot in the center of each burger. Put the burgers in the skillet with the bacon fat, divot-side up (see Cook’s Note). Cook the burgers over medium-high heat, turning once, until golden brown on both sides and cooked to medium doneness, about 8 minutes total. Use a heatproof spoon to remove some of the bacon fat from the skillet as the burgers cook if desired.
Top each burger with 2 slices of the cheese. Carefully add 1/4 cup of water to the pan and immediately cover tightly with a lid. Count to 10. Lift the lid or dome to make sure that the cheese is totally melted (if not, cover again and count 5 seconds longer). Transfer the burgers to a plate and loosely tent with foil.
Meanwhile, melt 1 tablespoon of butter in 1 tablespoon of canola oil in a nonstick skillet over medium-high heat. Carefully crack 2 eggs into two small bowls. Add the eggs to the pan, season with salt and pepper and reduce the heat to medium. Cover the skillet with a lid for 1 to 2 minutes to "baste"; the yolk should still be runny. Remove the eggs from the pan and set aside. Repeat with the remaining 1 tablespoon of butter, 1 tablespoon of canola oil and 2 eggs.
Stir together the ketchup and hot sauce in a small bowl.
Toast the buns in a toaster. Spread the spicy ketchup on the top bun. Place the burgers on the bottom buns and top each with a fried egg and 2 strips of crispy bacon. Close the burgers and serve with more spicy ketchup if desired.
The burgers can also be cooked in 2 tablespoons of canola oil instead of the bacon fat if desired.