For the cupcakes: Place an oven rack in the center of the oven and preheat to 350 degrees F. Line two 12-cup mini muffin tins 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 currants and grated carrot.
Drop 1 well-rounded tablespoon of batter into each paper liner. Bake until a toothpick inserted in the center of the cupcakes comes out clean, about 17 minutes. Cool the cupcakes in the tins for 5 minutes. Transfer to a rack and cool completely.
For 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 each cupcake with 1 heaped teaspoon of frosting and serve.