In 10-inch omelet pan or skillet with oven-proof handle* over medium-high heat, heat butter until just hot enough to sizzle a drop of water. Pour in egg mixture. Gently smooth surface. Reduce heat to medium. Cook until puffed and lightly browned on bottom, about 5 minutes. (Lift omelet at edge to judge color.)
Bake in preheated 350°F (175°C) oven until knife inserted halfway between center and outer edge comes out clean, about 10 to 12 minutes.
Loosen omelet edges with spatula. With sharp knife, cut upper surface down center of omelet but DO NOT cut through to bottom of omelet. Arrange reserved vegetable mixture over half of omelet. Sprinkle with cheese. Tip pan. With pancake turner, fold omelet in half and invert onto warmed plate or platter with a quick flip of the wrist. Cut in half or into wedges. Serve immediately.