Add oil to a heated saute pan. When hot, add onions and caramelize until nice and brown. Set aside. Break the eggs into a mixing bowl, add water, salt and pepper. Melt the butter in an 8-inch non-stick pan and set over high heat. Make sure the butter coats the whole pan. When the foam has subsided, slowly pour in the eggs. Tilt the pan to spread the eggs mixture evenly. Let eggs firm up a little and then shake the pan and use the spatula to get the mixture away from the sides, into the middle, letting the liquid run under the sides. Continue to cook until the mixture holds together. Add mustard, caramelized onions and grated cheese to the center. Tilt the pan to one side, and with a spatula fold about 1/3 of the omelet over the middle. Shake the pan gently to slide the omelet to the edge of the pan. Turn the omelet directly onto the plate by inverting the pan over the plate.