Heat 1/2 teaspoon oil in a medium nonstick skillet coated with cooking spray over medium-high heat. Add mushrooms, asparagus, and green onions; sauté 5 minutes or until crisp-tender. Remove from pan, set aside, and keep warm.
Combine eggs, egg white, and next 3 ingredients in a bowl, stirring well with a whisk. Heat remaining 1/2 teaspoon oil in pan. Add egg mixture to pan; cook until edges begin to set (about 2 minutes). Gently lift edges of omelet with a spatula, and tilt pan so uncooked portion flows underneath. Continue cooking until center is just set (about 4 minutes).
Spoon vegetable mixture evenly over half of omelet, and sprinkle cheese over vegetable mixture. Loosen omelet with spatula, and fold in half. Cut omelet in half crosswise, and serve immediately.