Preheat oven to 350. Set paper cupcake liners in 2 muffin pans to make room for 24 cupcakes (my muffin pans have 12 spots each). In an electric stand mixer, combine sugar, oil, and applesauce. Add eggs one at a time. Then add dry ingredients and mix to combine. Add carrots, nuts, and cherries, and mix again. Fill each cupcake liner with about 1/2 cup of batter.
Place on middle rack of oven and bake until set, about 45 minutes. Remove from the oven and allow cupcakes to cool.
Combine all ingredients in electric stand mixer and blend for several minutes until all lumps of powdered sugar disappear.
You can frost the cupcakes by dunking them upside down into the frosting, twisting, and flicking a cute little peak over on top when you pull them out.
I like to put a dried cherry or a sliver of crystallized ginger on top.