In a large bowl, beat the sugar, oil and egg until well blended. Beat in applesauce and vanilla. Combine the flour, baking soda, cinnamon and salt; gradually beat into sugar mixture until blended. Stir in carrots.
Fill paper-lined muffin cups half full. Bake at 350° for 15-20 minutes or until a toothpick inserted in the center comes out clean. Cool for 10 minutes before removing from pan to a wire rack to cool completely.
In a small bowl, combine frosting ingredients; beat until smooth. Frost cupcakes. Refrigerate leftovers.
Decorative cupcake liners can be found at fancyflours.com.