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Mash banana in a bowl with a fork or potato masher.
Whisk together all of the dry ingredients (flour, baking powder + soda, sugar, salt) in a large bowl.
Carefully separate egg yolk and egg white. Then add the yolk to the dry ingredients.
Add all of the other wet ingredients (applesauce, vanilla, oil, banana).
And stir with a spatula until well combined.
Beat the egg white until you can see stiff peaks forming.
Carefully fold in whipped egg whites.
Fold in walnuts.
Transfer the muffin dough to a muffin tin lined with muffin liners and bake for 15 minutes.
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