Toast the English muffin halves in a toaster or skillet until golden and crisp.
Heat 1 tablespoon butter in a nonstick skillet over medium heat until melted and foaming.
Crack the eggs into the skillet, season with salt and black pepper, and cook for 2 to 3 minutes until whites are set, then flip and cook for 1 minute more.
Place one cheese slice on each egg during the last 30 seconds of cooking to melt.
Spread the remaining butter on the toasted muffin halves, then place one egg with melted cheese on each bottom half.
Top with the remaining muffin halves and serve immediately while warm.