Whisk eggs in a bowl; add milk and season with salt and white pepper. Whisk until egg mixture is foamy, 2 to 3 minutes.

Melt butter in a small, nonstick skillet over medium-low heat. Pour in egg mixture and swirl the skillet, so the bottom is evenly covered with egg. Cook until egg starts to set, about 1 minute. Lift edges with a spatula and tilt the skillet so uncooked egg can flow towards the bottom of the skillet to set. Repeat until no visible liquid egg remains.

Carefully flip omelette; cook until warmed through, 30 seconds to 1 minute. Sprinkle Emmentaler cheese down the middle of omelette; fold in half. Cook until cheese is melted, about 20 seconds. Slide omelette onto a plate.