Preheat the oven to 375 degrees F. Lay the bacon slices on a rack set over a baking sheet. Bake until the fat starts to render, about 10 minutes.
Baste the bacon on both sides with the maple syrup and continue to bake until caramelized and the edges just start to brown, 10 to 15 minutes more. Let cool on the rack, about 5 minutes. When cool, cut them in half.
Melt 1 tablespoon of the butter in a large nonstick skillet over medium heat. Crack the eggs into the skillet and fry until thoroughly set, 3 to 4 minutes. Transfer to a plate.
Put 4 slices of the bread on a work surface. Layer each with 1 whole slice of the cheese, 3 pieces of the bacon, 1/4 slice of the cheese and 1 egg and another whole slice of cheese. Close the sandwiches with the remaining bread. Thinly and evenly spread the remaining butter on both sides of the bread using about 1 1/2 tablespoons for each sandwich.
Place 2 sandwiches into the same skillet over medium heat. Cover the skillet with a lid and cook until the bread is evenly golden brown, about 2 minutes. Flip and cook, uncovered, until the bread is golden brown and cheese is visibly melted, about 2 minutes more. Repeat with the remaining sandwiches.