Cut a hole in the center of each slice of bread with a round cookie cutter or the rim of a glass.
Place the bread into the hot skillet, crack one egg into the hole of each slice of bread, break the yolk and cook until the white is mostly opaque. Flip the slices of bread over. Form sandwich halves by topping each piece of bread with 1 slice of cheese, a slice of ham, then another slice of cheese. Season with salt and pepper and cook until the eggs are fully set; remove from the skillet.
Make two sandwiches by putting together the halves.
Place on a microwave-safe plate and microwave on high until the cheese is fully melted, 30 to 45 seconds.
