In a large bowl, beat the sugar, oil, egg whites and egg until well blended. Add applesauce and molasses; mix well. In a small bowl, combine the flours, baking soda, ginger, cinnamon, allspice and salt; gradually beat into applesauce mixture until blended.
Fill paper-lined muffin cups two-thirds full. Bake at 350° for 18-22 minutes or until a toothpick inserted in the center comes out clean. Cool for 10 minutes before removing from pans to wire racks to cool completely.
Just before serving, top each cupcake with 1 tablespoon whipped topping.