Using a small empty can, a biscuit cutter, or the ring from a Mason jar, cut a circle out of the center of each slice of bread.

In a large nonstick or cast iron skillet, melt 1 tablespoon butter over medium heat until foaming subsides. Add both bread slices and circular cutouts and cook until very lightly browned on the bottom side.

Break an egg into each hole, season with salt and pepper, and cook, swirling occasionally, until bread is lightly golden brown on the bottom side and eggs can easily be lifted without breaking, 1 to 2 minutes. Flip circular cutouts as soon as they are golden brown on the first side, then toast on the second side and remove when ready.

Add remaining 2 tablespoons butter and carefully flip both slices of bread, making sure the eggs don't break. Add a cheese slice to the top of each bread slice.

Cook, swirling the pan occasionally, until golden brown on the second side.

Close the sandwich with cheese inside and continue to cook, flipping occasionally, until eggs are soft-cooked and cheese is melted, about 1 minute longer. Serve immediately.