Cook the bacon in a medium skillet over medium heat until crisp, about 6 minutes. Transfer to a paper towel-lined plate, leaving about 1 teaspoon of fat in the pan.

Spread butter on the cut sides of the muffins and toast in the skillet until golden, about 2 minutes per side. Remove them.

Reduce heat to medium-low, crack the eggs into the skillet, and season with salt and pepper. Cook until whites are set but yolks remain runny, about 3 minutes.

Build the sandwiches: bottom muffin, cheese slice, folded bacon, fried egg, and top muffin. Press gently and serve right away.