In a medium bowl, whisk together the eggs, milk, salt, and pepper until well combined and slightly frothy.
Heat a non-stick skillet over medium heat and add the butter, allowing it to melt and coat the bottom of the pan.
Add the chopped spinach to the skillet and sauté for about 1-2 minutes until wilted.
Pour the egg mixture over the spinach in the skillet, tilting the pan to ensure even coverage.
Cook for 2-3 minutes, or until the edges begin to set.
Sprinkle the shredded cheese evenly over one half of the omelette.
Carefully fold the other half of the omelette over the cheese and cook for an additional 1-2 minutes until the cheese is melted and the eggs are fully cooked.
Slide the omelette onto a plate and serve immediately.