Preheat oven to 350°.
Combine oil and sugars in a large bowl; beat with a mixer at medium speed until well blended. Add egg and egg white; beat well. Beat in vanilla and applesauce. Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour and next 7 ingredients, stirring well with a whisk. Add flour mixture to applesauce mixture.
Pour batter into an 8-inch square baking pan coated with cooking spray. Bake at 350° for 22 minutes or until a wooden pick inserted in center comes out clean. Cool in pan 10 minutes.
Spread top of cake with Cream Cheese Frosting, and sprinkle with walnuts. Cut into 9 equal squares.