Preheat oven to 375°F. Lightly grease muffins cups or use muffin liners.
In a large bowl, mix together the flours, baking soda, baking powder, salt, cinnamon, & walnuts.
In another bowl, beat together bananas, sugars, egg, oil, & applesauce.
Stir the banana mixture, into the flour mixture, just until moistened.
Spoon batter into prepared muffin cups.
In a small bowl, mix together 1/3 cup brown sugar, 2 Tbs. flour, & 1/4 tsp. cinnamon. Cut in 1 Tb. butter, until mixture resembles coarse cornmeal. Add walnuts. If mixture seems too dry, add part or all of the 1/2 tsp. of butter Sprinkle topping over muffins. Push topping down a little on each muffin, so it will stay better.
Bake in the preheated oven for 18 to 20 minutes, until a toothpick inserted in center of a muffin comes out clean. Take out of oven.
Cool slightly in pan on rack; approx. 3 minutes. Then, take the muffins out of the pan & place on rack for further cooling.