Spread 1 side of each slice of bread with the mayonnaise. Top 4 slices of the bread with 1 slice of the American cheese and 1 slice of the Muenster cheese; top with the remaining 4 slices of bread. Cut the center out of each sandwich with a 2 1/2-inch ring mold.
Melt 2 tablespoons of the butter in a large nonstick skillet over medium-low heat and swirl to coat the bottom. Arrange the inner and outer parts of 2 of the sandwiches in the skillet, and cook until golden brown on the bottom, 4 to 5 minutes. Turn each over, crack an egg into each hole, sprinkle the egg with salt and pepper and cook until the bottoms of the sandwiches are golden brown, the egg whites are set and the yolk is still runny, 4 to 5 minutes. For the last minute of cooking, cover the pan with a lid to cook the egg white completely through. (The round sandwiches may finish cooking faster.) Transfer the egg sandwiches and the round sandwiches to serving plates and keep warm. Wipe out the skillet and melt the remaining 2 tablespoons of butter and repeat with the remaining 2 sandwiches and 2 rounds.
Serve each egg sandwich along with a round sandwich to dip into the egg yolk.
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