Beat the eggs with the scallion and pepper in a bowl. Put the bread and cheese within reach. Warm the skillet over medium-low heat and melt the butter, swirling to coat the base.

Pour in the eggs. Immediately lay the bread slices side by side on the wet egg, then turn each slice once so both surfaces pick up a little egg. Leave a small gap between the slices.

Cook for about 2–3 minutes, until the egg sheet is firm enough to lift and the top is mostly set. Loosen around the edge. Slide a wide spatula under the bread and egg and carefully turn the whole sheet over; use a second spatula if helpful.

Fold any egg overhang inward over the bread. Put the cheese on one slice, then fold the other bread slice over it to close the sandwich. Toast gently for about 1–2 minutes per side, until golden, the cheese melts, and the egg center reaches 160°F/71°C. Reduce heat if the bread browns before the egg cooks; allow extra time if needed.

Transfer to a plate and cut in half. Serve immediately. Refrigerate leftovers within 2 hours (1 hour above 90°F/32°C), in a covered container at 40°F/4°C or colder, and use within 3–4 days. Reheat until the center reaches 165°F/74°C; the toast will be softer than when fresh.