Preheat broiler. In a broilerproof medium skillet, heat oil over medium heat. Add broccoli, mushrooms, carrot, and onion; cook for 7 to 8 minutes or until crisp-tender, stirring occasionally. (If vegetables start to overbrown, reduce heat.)

Meanwhile, in a medium bowl, whisk together egg whites, eggs, basil, salt, and pepper. Pour egg mixture over vegetables 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 and surface is just slightly moist. Sprinkle with cheese. Broil 4 inches from the heat about 2 minutes or until top is lightly browned and center is set. Let stand for 5 minutes before serving.

If desired, substitute 1 1/2 cups refrigerated or frozen egg product, thawed, for the egg whites and eggs.