Beat eggs in a bowl with a whisk. Add milk and season with salt and white pepper. Whisk for a few minutes until egg mixture is foamy; beating in air makes the omelette fluffy.
Melt butter in a small, nonstick skillet over medium-low heat. Pour in egg mixture and twirl skillet so bottom is evenly covered with egg. Cook for 1 minute until egg starts to set. Lift edges with a spatula and tilt the skillet so uncooked egg mixture can run towards the bottom of the skillet to set. Repeat until no visible liquid egg remains.
Carefully flip omelette and cook for another 30 seconds to 1 minute. Sprinkle Emmentaler cheese in one line in the middle of the omelette and fold omelette in half. Cook for 20 seconds, then slide omelette onto plate.