Heat a 12-inch skillet over medium-high heat. Add the ground beef, press it flat and leave it alone for 2 minutes, then break it into small crumbles and cook 4 to 5 minutes more until browned with no pink left.

Spoon off all but about 1 tablespoon of fat. Add the onion and bell pepper and cook 4 to 5 minutes, stirring, until the onion turns soft and translucent at the edges.

Stir in the garlic and smoked paprika and cook 30 seconds, just until fragrant.

Add the tomato sauce, ketchup, brown sugar, Worcestershire, mustard, salt and pepper. Stir well and lower the heat to medium-low.

Simmer 10 to 12 minutes, stirring now and then, until thick enough that a spoon dragged through it leaves a trail for 1 to 2 seconds. The beef should read 160°F (71°C). Taste and adjust salt.

While the filling simmers, spread 1 tablespoon butter on the cut sides of the buns and toast them cut side down in a nonstick skillet over medium heat for 1 to 2 minutes, until golden.

Melt the remaining 1 tablespoon butter in the same skillet over medium heat. Crack in the eggs (in batches if needed) and cook 3 to 4 minutes until the whites are set.

Lay a slice of cheese on each egg, cover the pan and cook 30 to 60 seconds until the cheese melts. For firm yolks, keep covered 1 to 2 minutes longer.

Spoon about 1/2 cup of filling onto each bottom bun, top with a cheesy egg and the top bun. Serve right away.