Prepare a muffin tin by spraying with cooking spray, 10 of the cups. Or, line 10 cups with silicone muffin cups.
Pour egg whites equally into the 10 cups. It will be a small amount in each cup. This makes your egg muffin thin, and easier to eat on a bagel! You don't want it too tall.
Sprinkle the egg muffins with fat free cheddar cheese, equally among all ten muffins.
Bake in oven at 425 for about 8-10 minutes, or until eggs are cooked.
While eggs are cooking, split each bagel and toast. Spread mashed avocado among all bagel halves.