Whisk together eggs, milk, 1/8 teaspoon salt and pepper to taste in a large bowl until well combined.
Melt butter over medium heat in a large sauté pan. When butter has melted but not yet browned, add egg mixture. After about 30 seconds, tilt pan while simltaneously using a spatula to push edges of omelet inward toward center of pan. (Tilting the pan will force the runny egg mixture to fill around edges.) Repeat, while rotating pan, until omelet is set and cooked through, 2 to 3 minutes.
Sprinkle Monterey Jack over half of omelet and top same half with avocado slices. Fold other half of omelet over. Sprinkle with salt and pepper to taste and cut into 4 equal pieces. Serve immediately, topped with dollops of sour cream, if desired.