1. Gently beat eggs and egg whites together in a medium bowl. Heat oil in a 10-inch nonstick skillet over medium-high heat. Add onion and 1/4 teaspoon each salt and pepper and cook, stirring occasionally, until edges begin to brown, about 10 minutes.
2. Reduce heat to medium low and cook onion 20 minutes more, or until golden and tender, adding 2 tablespoons water every 5 minutes and stirring occasionally. Transfer onion to a bowl, wipe skillet clean and raise heat to medium. Mist with cooking spray, add arugula and cook, stirring, until slightly wilted, 1 to 2 minutes. Transfer to a plate.
3. Wipe skillet clean and mist with cooking spray; heat over medium heat. Add 1/6 of the egg mixture. Cook 15 seconds, stir with a silicon spatula, then swirl to evenly recoat pan. Cook until just set and no longer wet, about 2 minutes.
4. Sprinkle omelet with 1/8 teaspoon each salt and pepper. Arrange 2 tablespoons caramelized onions, 1/4 cup wilted arugula and 2 tablespoons goat cheese down center of omelet. Gently fold omelet into thirds over filling and slide onto a plate. Garnish with fresh arugula. Repeat with remaining ingredients.