Spread 1 side of each slice of bread with mayonnaise. Top 1 slice with American cheese and the other with pepper Jack, then sandwich them together. Cut the center out of the sandwich with a 2 1/2-inch ring mold.
Crack the egg into a small bowl.
Melt the butter in a medium nonstick skillet over medium-low heat, and swirl to coat the bottom. Arrange the inner and outer parts of the sandwich in the skillet, and cook until golden brown on the bottom, 4 to 5 minutes. Turn each over, pour the egg into the hole, sprinkle with salt and pepper and cook until the bottoms of the bread 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 sandwich round may finish cooking faster.) Transfer the sandwich with the egg to a serving plate along with the sandwich round, to dip into the egg yolk.
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