To prepare the cake, spoon the applesauce into a fine sieve over a bowl; let stand 15 minutes. Discard liquid. Scrape drained applesauce into a bowl; cover and refrigerate.
Combine carrot and granulated sugar in a colander. Drain 20 minutes.
Beat the brown sugar and butter with a mixer at medium speed until well-blended (about 2 minutes). Add applesauce, 1 teaspoon vanilla, and coconut extract; beat until well-blended (about 2 minutes). Add eggs and egg whites, 1 at a time, beating well after each addition (batter will have a slightly curdled look).
Lightly spoon flour into dry measuring cups, and level with a knife. Combine flour and next 5 ingredients (flour through salt); stir well with a whisk. Add brown sugar mixture to flour mixture, stirring just until moist. Fold in carrot mixture and raisins (batter will be very thick). Spoon batter into a 13 x 9-inch baking pan coated with cooking spray.
Bake at 350° for 40 minutes or until a wooden pick inserted in center of cake comes out clean. Cool completely on a wire rack.
To prepare the frosting, beat the powdered sugar, 2 teaspoons vanilla extract, and cheeses with a mixer at low speed just until well-blended (do not overbeat). Spread the frosting over the cake; cover and chill 1 hour.