Heat a long griddle over two stovetop burners on medium heat. Spray with cooking spray and when hot, pour half of the egg whites on, season with salt and pepper.
When the eggs appear to be cooked (about 2-3 minutes), use a spatula to carefully fold the eggs over themselves. Fold over enough times so that it will fit on the bread.
Top the finished egg with half of the shredded cheese and melt. Spread the toast on both sides with half of the avocado and season with salt and pepper. When the cheese has melted, add the egg to the toast.