To make the cake, preheat oven to 350F. Lightly coat a 9-by-13-inch baking pan with nonstick cooking spray. In a bowl, combine flour, sugar, cocoa powder and baking soda. Set aside. In a large bowl, using an electric mixer on the highest speed, beat the egg whites for 90 seconds. Add applesauce, buttermilk and vanilla. Beat until smooth. Add half of the flour mixture and beat on a low setting until just mixed. Add the remaining flour mixture and beat on high for 2 minutes. Transfer batter to the prepared pan. Bake for 35-40 minutes, or until toothpick inserted in the center comes out clean and the sides of the cake begin to pull away from the pan. Remove from oven and cool completely.
To make the frosting, beat the butter in a small bowl until smooth and fluffy. Add 1/2 cup of powdered sugar; beat on low until smooth.
Add 1 tablespoon each of buttermilk and cocoa powder; beat until smooth. Add half of the remaining sugar, buttermilk and cocoa powder; beat well. Add remaining ingredients and beat until smooth. Spread over cooled cake. Scatter nuts evenly on top of the frosting. Slice and serve.