1. Place the two tablespoons of butter in a medium to large skillet and turn the heat to medium-high. Beat together the eggs and milk, just until blended. Add salt and pepper to taste.
2. When the butter melts, swirl it around the pan until its foam subsides, then pour in the egg mixture. Cook undisturbed for about 30 seconds, then use a fork or spatula to push the edges of the eggs toward the center. As you do this, tip the pan to allow the uncooked eggs in the center to reach the perimeter.
3. Repeat until the omelet is still moist but no longer runny, a total of about 3 minutes. Use a large spatula to fold the omelet in half or in thirds and place it on a plate. Rub to top of the omelet with the remaining 1 teaspoon of butter.