Preheat oven to 350°F. Grease two 8x5 inch loaf pans with baking spray.
In a medium bowl, combine the dry ingredients; flour, baking soda and salt with a wire whisk. Set aside.
In a large bowl cream butter and sugar with an electric mixer or stand mixer. Add egg whites, apple sauce and vanilla, and beat at medium speed until thick. Scrape down sides of the bowl.
Add 1/3 of the dry ingredients and blend at low speed until combined, add another 1/3 of the dry ingredients and blend at low speed, add remaining 1/3 of dry ingredients and the mashed bananas and blend at low speed until batter is combined. Do not over mix. Stir in the walnuts.
Pour the batter into loaf pans and bake on the center rack for 50 to 52 minutes, or until a toothpick inserted in the center comes out clean.
Let the pan cool at least 20 minutes, bread should be room temperature before slicing. Makes 2 loaves.