Heat 1/2 teaspoon oil in a small sauté pan over medium heat. Pour in the egg mixture, tilting the pan to spread it evenly across the bottom.
Allow the egg to cook for one minute, using a rubber spatula to gently pull the egg away from the sides of the pan so the rest of the liquid egg can continue to coat the pan and cook.
Using the spatula, fold the outside parts of the omelet inwards into a square shape (to fit the bread).
Remove the square-shaped omelet to a plate, place a slice of bread topped with 2 tablespoons of the shredded cheese, cheese side up, in the pan. Top with the omelette, another 2 tablespoons of cheese and slice of bread. Cook for 1 minute on each side until golden.