In a medium bowl combine eggs, basil, salt, and black pepper; set aside. Heat oil in a large broilerproof skillet; add vegetables and green onions. Cook, uncovered, over medium heat about 5 minutes or until vegetables are crisp-tender, stirring occasionally. Stir in meat.
Pour egg mixture over vegetable mixture in skillet. Cook over medium heat. As mixture sets, run a spatula around edge of skillet, lifting egg mixture so uncooked portion flows underneath. Continue cooking and lifting edge until egg mixture is almost set (surface will be moist). Sprinkle with cheese.
Preheat broiler. Place skillet under the broiler 4 to 5 inches from heat. Broil for 1 to 2 minutes or until top is just set and cheese is melted. (Or bake in a 400 degrees F oven for 5 minutes until top is set.)