In a medium bowl, crack in the eggs. Add the milk, kosher salt, and black pepper. Whisk together with a fork or small whisk until the mixture is smooth, the yolks and whites are fully combined, and slightly frothy on top.
Place an 8-inch nonstick skillet over medium-low heat. Add the butter to the skillet and let it melt, tilting the pan to coat the bottom evenly. The butter should bubble gently but not brown; if it sizzles too much, reduce the heat slightly.
Pour the egg mixture into the skillet. Let it sit undisturbed for about 30 seconds, just until the edges begin to set. Using a rubber spatula, gently stir the eggs in a circular motion, pulling the edges toward the center and tilting the pan to let the uncooked egg flow to the edges. Do this for about 30 seconds to 1 minute until the bottom is mostly set and the top is still just a bit glossy.
Once the eggs are mostly set, use the spatula to gently smooth the surface. Sprinkle the shredded cheese evenly over one half of the omelette.
Let the omelette cook for another 15 to 30 seconds, just until the cheese begins to melt and the top looks barely set.
Use the spatula to gently fold the omelette in half over the cheese side. Let it sit for another 10 to 15 seconds to help the cheese melt fully.
Slide the omelette onto a plate, garnish with fresh chives, and serve hot.