Whisk the eggs with the salt and pepper until no streaks of white remain.

Melt the butter in a small nonstick skillet over medium heat until it foams.

Pour in the eggs and swirl the pan to coat the bottom. As the edges set, pull them toward the center with a spatula and tilt the pan so the runny egg fills the gaps. Cook 1–2 minutes, until mostly set but still moist on top.

Sprinkle the cheddar over one half. Take the pan off the heat and let the cheese melt, about 20 seconds.

Fold the omelette in half and slide it onto a plate.