Cook the onion over medium heat with the coconut oil and salt until it begins to soften.
Once the onions have cooked down and begun to brown, add the apples. Once the apples are soft, reduce the heat to medium-low and continue to cook for about an hour. (Until the apples and onions are very soft and sticky. They should reduce in size.)
Beat the eggs and water together in a small bowl. Add a pinch each of salt and pepper.
Heat the butter or coconut oil in a 10-inch pan over medium-high heat. Pour in the eggs. Immediately, push the edges into the center and tilt the pan to allow the uncooked eggs to touch the pan and set up.
Once the top of the eggs has set up completely, spoon the caramelized apples and onions onto one half of the omelet. Add cheese, if desired. Fold the omelet in half and serve immediately.