In a small bowl, whisk together egg and 1 teaspoon cold water; season with salt and pepper. Using the rim of a small glass, cut a hole in bread; discard cutout.
In a large nonstick skillet, heat butter over medium until foaming. Place bread in skillet; carefully pour egg mixture into hole (if egg leaks out of the bread, use a spatula to push it back). Sprinkle egg with cheese.
Cook until egg begins to set, 3 to 4 minutes; gently flip bread and egg. Continue cooking until egg is set, 1 to 2 minutes.