Preheat oven to 325 degrees. Spray an 8x8 square dish with nonstick cooking spray and set aside.
In a large bowl, combine all the dry ingredients, excluding the sugar. Stir to combine and set aside.
In a different bowl, whip sugar, oil, and applesauce together. Stir in eggs, vanilla and mashed banana. Slowly incorporate the dry ingredients into the wet until just combined. Pour into prepared dish and bake 25-30 minutes or until toothpick comes out clean when inserted. Cool completely.
To make the frosting, place butter in a small saucepan over medium heat. Just after the butter melts, start swirling the pan in small circles. After about 30 seconds or so, the butter should start foaming and will turn an amber color and smell nutty. Once this happens, remove from heat immediately and pour into a different container. Cool to room temperature.
Once butter has cooled, whip together with vanilla, powdered sugar and cream until smooth and creamy. Spread over cooled cake and top with chopped walnuts, if desired. Store in an airtight container until ready to serve.