Preheat the oven to 350 degrees. Spray a 9-by-5 inch loaf pan with nonstick cooking spray. In a medium bowl, whisk the flours, baking powder, baking soda, cinnamon and salt to combine. In another medium bowl, beat the mashed bananas, sugar, applesauce, egg whites, and vanilla until well mixed. Add the dry ingredients and whisk just until smooth. Stir in the walnuts. Transfer the batter to the loaf pan and smooth the top. Bake for 50 minutes to 1 hour, until the top is browned and the sides of the bread pull away from the sides of the pan. Remove from the oven and let stand for 10 minutes on a wire baking rack. Invert and unmold onto the rack, turn right side up, let cool completely.