Preheat the oven to 325°F. Spray a 9 by 5 inch loaf pan with cooking spray.
Peel the 4 bananas into a large bowl and mash with a fork. Add the ½ cup applesauce, ½ cup sour cream, 2 eggs, and 1 tablespoon vanilla extract and whisk until combined.
Whisk in the ½ cup granulated sugar and ½ cup brown sugar until fully incorporated.
Add the 1¾ cups all purpose flour, 1 teaspoon baking soda, 1 teaspoon baking powder, and ¼ teaspoon salt. Stir with a wooden spoon just until the flour disappears. The batter will look lumpy and that's correct. Do not overmix.
Fold in the 1 cup chopped walnuts with a few gentle turns.
Pour the batter into the prepared pan. Bake for 50 minutes to 1 hour, until an instant read thermometer inserted into the center reads 200 to 205°F, the top is domed and cracked, and the edges have pulled slightly from the pan. If the top browns before the center sets, tent loosely with foil.
Cool in the pan for 15 minutes, then turn out onto a wire rack. Let it cool completely, at least 2 hours, before slicing.