Preheat oven to 325 degrees F (165 degrees C). Line the bottom of a 9x5-inch nonstick loaf pan with parchment paper.
Beat brown sugar and oil together in a large bowl until well-blended. Add eggs, applesauce, and vanilla extract; mix well to combine. Stir in mashed bananas. Add whole wheat flour and baking soda; stir until incorporated. Fold walnuts into the batter.
Pour batter into the prepared loaf pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour. Cover loaf with a clean towel and cool on a wire rack for 30 minutes before slicing.