Mini Carrot-Apple Cupcakes

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Cupcakes:

1 1/4 cups all-purpose flour (6 to 6 1/2 ounces) 

1 1/2 teaspoons baking powder 

3/4 teaspoon ground cinnamon 

1/2 teaspoon fine salt 

1/8 teaspoon ground allspice 

6 tablespoons (3/4 stick) unsalted butter, at room temperature 

3/4 packed cup dark brown sugar 

2 large eggs, at room temperature 

1/4 cup unsweetened applesauce 

1/4 cup fresh unfiltered apple juice 

1/3 cup dried currants 

1 small carrot, peeled and finely grated 

Cream Cheese Frosting:

4 ounces cream cheese, at room temperature 

1/3 cup powdered sugar 

1/4 cup chilled heavy whipping cream