Preheat oven to 350 degrees. Butter a 9-by-5-by-3-inch loaf pan, and set aside. Whisk together the flours, flaxseed, salt, baking powder, and baking soda in a medium bowl, and set aside.
Put egg and egg white in the bowl of an electric mixer fitted with the paddle attachment, and mix on medium-low speed until well combined, about 2 minutes. Add melted butter, sugar, vanilla, and bananas, and mix until combined. Add the reserved flour mixture, and mix on low speed until well incorporated, about 10 seconds. Stir in walnuts.
Pour batter into buttered pan. Bake until golden brown and a cake tester inserted into the center comes out clean, about 35 minutes. Let cool slightly in pan on a wire rack. Invert bread to unmold; reinvert, and let cool completely on rack.