Preheat the oven to 180°C / 350°F.
Lightly beat the egg in a mixing bowl. Add the vegetable oil and mix a bit more.
Add the brown sugar, bananas, applesauce, and vanilla extract. Mix lightly together until all ingredients are blended together.
In another mixing bowl, add the whole wheat flour, oat bran, cinnamon, salt, baking soda and baking powder. Mix together.
Fold gently the dry mixture into the wet mixture. Be careful not to overmix. Add the crushed walnuts if you are adding them.
Either put a paper liner in each muffin cup or lightly oil each muffin cup.
Fill 2/3 of each muffin cup with the muffin batter. Put the muffin tin in the oven. Bake about 20 to 25 minutes or until the muffins are golden brown on top. Remove the muffin tin and let cool about 10 minutes before removing the muffins from the baking tin.