Heat a nonstick pan on medium heat. Once hot, add the oil. Crack the eggs into the pan. Cook until the whites of the eggs start to firm up, break the yolk if desired, and gently flip the eggs over and top with cheese. Cook through until the whites are completely firm and cheese is melted. (Cover with a lid to help speed up the cooking process)
Place the eggs and avocado onto the bread, cut in half, and serve with peach.