Preheat oven to 400F.
Cover a baking sheet with aluminum foil and arrange the bacon slices on the pan. Drizzle the maple syrup over the slices and sprinkle them with brown sugar. Bake until the bacon is cooked through and golden brown, 8-10 minutes. Allow to cool.
When the bacon has cooled enough to handle, place a large skillet over medium heat with the butter. On a work surface, lay down 4 slices of bread and cover each of them with 2 slices of cheese. Lay down four strips of bacon on each slice and top them all off with 2 more slices of cheese. Cap off each sandwich with the remaining bread slices and dip them all in the beaten egg (just like French toast). Slide the sandwiches into the skillet and cook until golden brown and the cheese has melted, 4-5 minutes per side. Serve warm.