Line two 12- cup mini muffin pans with mini baking papers.
Whisk the flour, baking powder, cinnamon, salt and allspice in a medium bowl to blend.
Using a handheld electric mixer, beat the butter in a large bowl until fluffy.
Add the brown sugar and beat until blended. Beat in the eggs, one at a time.
Beat in half of the dry ingredients, then the applesauce and apple juice.
Beat in the remaining dry ingredients. Fold in the grated carrot.
Drop 1 well-round Tablespoon of batter into each paper liner.
Bake until a toothpick inserted into the center of the cupcake comes out clean, about 17 minutes. Cool the cupcakes in the tins for 5 minutes. Transfer to a rack and cool completely.
Frosting
To make the frosting, combine the cream cheese, powdered sugar, and whipping cream in a deep medium bowl. Beat until the frosting is thick and smooth. Cover and chill for at least 1 hour until firm. Frost cupcakes.