Heat 1 1/2 teaspoons butter in a skillet over medium heat; arrange bagel halves, cut sides down, in the melted butter. Cook until bagel is very slightly toasted and softened, 1 to 2 minutes per side. Transfer bagel to a plate.
Cook ham slices in the same skillet until heated through, 1 to 2 minutes. Arrange 2 ham slices onto each bagel half.
Heat remaining 1 1/2 teaspoons butter in the same skillet; cook eggs in the melted butter until whites are set but yolks are still runny, about 3 minutes. Gently flip eggs and cook until whites are set, about 1 more minute. Season eggs with salt and pepper. Layer eggs and American cheese slices onto 1 bagel half; top with the second bagel half, creating a sandwich.
Place bagel sandwich back in the skillet and cover with a lid; heat until cheese is melted, about 45 seconds per side.
Transfer bagel sandwich to a plate; drizzle with maple syrup.