In a mixing bowl, crack the eggs and add the milk. Add salt and black pepper to taste. Whisk until the mixture is well combined and slightly frothy.
Heat a non-stick skillet over medium heat and add the butter. Allow it to melt and coat the bottom of the skillet.
Pour the egg mixture into the skillet, tilting it to ensure the eggs spread evenly across the surface.
Cook for about 2-3 minutes until the edges begin to set. Gently lift the edges with a spatula, letting any uncooked egg flow underneath.
Once the eggs are mostly cooked but still slightly runny on top, sprinkle the shredded cheese over one half of the omelette.
Carefully fold the other half of the omelette over the cheese. Cook for an additional 1-2 minutes until the cheese has melted and the omelette is firm and cooked through.
Slide the omelette onto a plate, cut it in half, and serve warm. Garnish with chopped herbs if desired.