Whisk eggs, egg whites, milk, pumpkin pie spice, vanilla and pumpkin in bowl until frothy.
Combine apples and bread in separate bowl. Pour egg mixture on top, stir and let sit 5 minutes. Press mixture evenly into muffin tin.
Bake 45 minutes, or until tops begin to brown and are cooked through. Immediately run knife around edges of muffins to loosen and remove from muffin tins.