Cook the bacon in a large skillet over medium heat until crispy. Transfer to a paper towel-lined plate to drain.
Spread some of the butter on one side of each slice of sourdough bread. Set aside.
Melt the remaining butter in a clean skillet over medium-high heat.
Crack the eggs into the skillet and season with salt and pepper. Cook for 2–3 minutes, until the bottoms are set. Flip and cook for 1 additional minute.
Place the cheddar cheese slices over the eggs and cook until the cheese begins to melt. Transfer the eggs to a plate and set aside.
Place 2 slices of bread, butter-side down, in the skillet. Top each with 1 slice of American cheese.
Top each sandwich with 2 eggs and 2 slices of bacon. Add the remaining bread slices, butter-side up.
Cook for 3–4 minutes, until the bottoms are golden brown. Flip and cook for another 3–4 minutes, until the second side is golden brown and the cheese is melted.
Remove the sandwiches from the skillet, cut in half, and serve immediately.
Notes
These sandwiches are best served fresh. Mayonnaise can be used instead of butter on the bread, if preferred. You can also substitute your favorite sliced cheese.