Preheat oven to 350 degrees.
Move oven rack to low position so tops of pans will be in center of oven.
Prepare all-purpose flour. Place one large and one medium bowl on counter. Place sifter in large bowl. Pour 3 cups all-purpose flour into the sifter (this is more than recipe calls for). Sift flour into the large bowl. Discard particles left in the sifter.
Lightly spoon sifted flour into measuring cup until it's piled high. Level top with smooth end of knife. Pour into medium bowl. Repeat until you have a total of 2-2-1/2 cups of flour (sifted, spooned and leveled).
Return excess flour to the container. Wipe large bowl clean with a paper towel and set aside to use later.
Add baking soda and salt in the bowl with the all-purpose flour. Whisk together until well combined. Set aside.
In large bowl, cream together sugar and softened butter. Add applesauce and two eggs. Stir until well combined. Add bananas, buttermilk and vanilla extract. Beat until smooth (small lumps okay).
Add prepared all-purpose flour mixture (with baking soda and salt). Stir just until moistened. Do not over mix. Gently stir in chopped walnuts (if using).
Grease bottoms only of 2 loaf pans, 8 1/2x4 1/2x2 1/2 inches or 1 loaf pan, 9x5x3 inches. Pour batter into prepared pan(s).
Bake 8-inch loaves about 1 hour, 9-inch loaf about 1 1/4 hours OR until a long wooden skewer or butter knife inserted in center and gently touches the bottom... comes out clean. Toothpick is too short because it doesn't pierce through to the bottom.
Cool in pan for 10 minutes. Gently loosen sides of loaves from pans. Remove and place top side up on wire rack. Cool completely (about 2 hours) before slicing.
Wrap tightly and store at room temperature up to 4 days. Or refrigerate for up to 10 days.