Check that the two bread slices fit side by side in a 12-inch (30 cm) nonstick skillet with a little room between them. Beat the eggs and pepper in a bowl until the whites and yolks are evenly mixed.

Warm the skillet over medium-low heat. Melt the butter and swirl to coat. Pour in the eggs and tilt into a thin, even layer.

Immediately lay both bread slices side by side on the wet egg, briefly coat one side, then turn each slice over so both sides have some egg. Leave a narrow gap between the slices. Cook for 2–3 minutes, until the egg forms one set sheet that releases with a spatula.

Slide a wide spatula beneath the egg-and-bread assembly, support it with a second spatula if needed, and turn it over as one piece so the bread touches the skillet. Fold the excess cooked egg edges over the bread.

Place the cheddar over one slice. Fold the other slice over it to close the sandwich. Cook gently for about 2 minutes per side, pressing lightly, until both bread faces are golden, the cheese melts, and the egg center reaches 160°F (71°C). Reduce the heat if the bread colors too quickly.

Transfer to a board, rest for 1 minute, and cut in half. Eat while the bread is crisp.