Crack the eggs into a mixing bowl. Add the milk, salt, and pepper. Whisk until bubbles appear and the yolks and whites are fully combined.

Place the nonstick skillet on medium-low heat. Add the butter and let it melt, swirling to coat the pan.

Pour the egg mixture into the skillet. Let it sit undisturbed for 20 seconds until the edges start to set.

With the spatula, gently pull the eggs from the edges toward the center, forming soft curds.

When the eggs are half-cooked but still a little runny, sprinkle the shredded cheese evenly over the eggs.

Continue gently stirring and folding the eggs until just set and the cheese is melted, about 1 to 2 minutes more.

Remove from heat immediately to prevent overcooking. Divide onto two plates and serve hot.