Preheat oven to 400º F. Peel and core apple. Thinly slice apple and onion.
Beat eggs with milk, water, pepper and nutmeg in a small bowl; set aside.
Over medium heat, melt butter in a small, ovenproof skillet. Add onion and apple to the skillet and sauté until onion becomes translucent, about 5 to 6 minutes.
Spread out the onion and apple mixture evenly in the skillet. Pour egg mixture evenly into the skillet and cook over medium heat until the edges begin to set. Sprinkle the cheddar cheese over the top. Transfer skillet to the oven and bake until the center is firmly set, approximately 10 to 12 minutes.
Cut omelet in half and slide each half onto an individual plate. Serve immediately.