Preheat broiler. In a bowl whisk together eggs, milk, herb, the 1/2 teaspoon salt, and the 1/4 teaspoon black pepper; set aside.
In a large broilerproof skillet, heat butter over medium heat. Add vegetables; cook and stir for about 3 minutes or until crisp-tender. Stir in cooked meat; cook and stir for another 1 to 2 minutes or until heated through. Season mixture with additional salt and black pepper.
Pour egg mixture over the vegetable-meat mixture in the skillet. Cook over medium heat. As the egg mixture sets, run a spatula around edge of skillet, lifting egg mixture so uncooked portion flows underneath. Continue cooking and lifting edges until egg mixture is almost set (surface with be moist). Reduce heat as necessary to prevent overcooking. Sprinkle with cheese.
Broil 4 to 5 inches from the heat for 1 to 3 minutes or until top is set and cheese is melted. Makes 4 servings.