On a cutting board, place one half of the bagel cut-side down. Layer the American and Muenster cheeses over the bagel half, alternating slices. Top with the other half of the bagel, seed-side down, and press lightly.
In a small skillet, melt 2 tablespoons of the butter over medium heat. Cook the bagel, weighed with a sandwich press or another clean skillet, until the bottom side is golden brown, 2 to 3 minutes. Flip the sandwich, adding the remaining 1 tablespoon of butter, and cook until golden brown and the cheese is melted, 2 to 3 minutes more. Transfer to a cutting board and slice in half, then serve.