Heat remaining 2 teaspoons oil in same skillet over medium-low heat. Pour egg mixture into skillet. Sprinkle with cheese. Cover skillet and cook until eggs are set on bottom and cheese melts, about 10 minutes. Place skillet under broiler and cook just until eggs are set on top and cheese bubbles, about 2 minutes.
Transfer omelet to platter.
Cut into wedges and serve with a chilled bottle of Carta Nevada Sparkling Wine.