Combine first 8 ingredients in a bowl, stirring with a whisk.
Melt butter in a 9-inch nonstick skillet over medium heat. Pour egg mixture into pan. As mixture starts to cook, gently lift edges of omelet with a spatula, and tilt pan so uncooked portion flows underneath. Cook just until set (about 5 minutes).
Sprinkle cheese over omelet. Fold omelet in half; cover and reduce heat to low. Cook 2 minutes. Cut omelet in half crosswise. Serve immediately.