Heat a nonstick or well-seasoned skillet over medium heat and melt about 1 teaspoon of the butter. Crack in the egg, season it with salt and pepper, and cook for about 2 to 3 minutes, until the white is set and the yolk reaches your preferred doneness. Flip briefly if you prefer a fully cooked yolk, then transfer the egg to a plate.

Butter one side of each bread slice with the remaining butter. For extra browning, spread a thin layer of mayonnaise on the outside instead of, or together with, a small amount of butter.

Place one bread slice buttered side down in the skillet. Add the cheese and fried egg, then top with the second bread slice, buttered side up.

Cook over medium-low heat for 2 to 3 minutes, until the underside is golden brown and the cheese is beginning to melt. Carefully turn the sandwich and cook for another 2 to 3 minutes, pressing gently with a spatula without flattening it heavily.

Remove the sandwich when both sides are crisp and the cheese is fully melted. Rest for 1 minute, then cut and serve immediately.