Crack the eggs into a mixing bowl. Add milk, salt, and black pepper. Whisk until well combined and slightly foamy.

Dice the ham and shred the cheese if not already prepared.

Heat the nonstick pan over medium heat. Add butter and let it melt, swirling to coat the bottom.

Pour the egg mixture into the pan. Let it cook undisturbed for 1 minute, then gently pull the cooked edges toward the center with a spatula, allowing uncooked egg to flow to the edges.

When the eggs are mostly set but slightly runny on top (about 2-3 minutes), sprinkle diced ham and shredded cheese evenly over one half of the omelette.

With the spatula, carefully fold the other half over the filling. Cook for 1-2 more minutes until the cheese melts and the eggs are cooked through.

Slide the omelette onto a plate. Garnish with chopped chives or parsley if desired.

Serve hot.