Finely dice the red pepper and season with salt and pepper.
Butter one side of each slice of bread. Spread half of the chopped red pepper on the other side of each bread slice, laying the slice butter-side down. Top one bread slice with the cheese, then sandwich it all together. Using a round biscuit cutter, cut out the center of the sandwich.
Heat a nonstick skillet over medium heat. Once hot, melt some butter in the skillet. Add the sandwich and its cut-out center to the pan. Crack an egg into the center of the sandwich and let cook for 3-4 minutes; flip the sandwich and cut-out center and cook a few minutes more or until egg is cooked to desired doneness.