Preheat the oven to 350 °F (175 °C) and prepare a 9-by-5 inch loaf pan by spraying the inside of the pan with cooking spray.
Remove the banana skin and place the bananas in a mixing bowl. Mash the bananas using a potato masher. Add the brown sugar to the mashed bananas and mix until smooth using a whisk or a fork.
Whisk the beaten eggs, applesauce, and vanilla extract into the banana mixture.
In a separate bowl, add the flour, baking powder, baking soda, and salt. Mix the ingredients together.
Pour the dry ingredients into the wet ingredients and fold together using a spatula until no dry flour mixture remains.
Gently fold ¾ a cup of the walnuts into the banana mixture.
Pour the batter into the prepared loaf pan and sprinkle the remaining walnuts over the top of the batter.
Bake the loaf for 50 minutes at 350 °F (175 °C) in the oven or until a toothpick inserted at the centre of the loaf comes out clean.
Remove the loaf from the oven and let it cool in the pan for at least 5 minutes. Once the loaf has cooled down enough to safely handle, remove it from the pan using a spatula and let it cool completely on a wire rack before slicing and serving.