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1. Preheat oven to 350 degrees F. Line twenty-four 2 1/2-inch muffin cups with folded 5x4-inch rectangles of parchment paper or paper bake cups; set aside.
2. In a medium bowl, combine the flour, pumpkin pie spice, baking powder, baking soda, and salt; set aside. In a large mixing bowl, beat the butter with electric mixer on medium speed for 30 seconds. Add granulated sugar; beat until light and fluffy, about 2 minutes. With mixer on low, beat in eggs one at a time, stopping to scrape down sides of bowl between additions. Add the sweet potato, milk, and vanilla; beat on low speed until combined. Add the flour mixture to the sugar-egg mixture; beat on low just until combined.
3. Fill each muffin cup about two-thirds full with batter. Bake for 5 minutes. Carefully remove pan from oven. Gently press kisses, tips up, about halfway into each cup. Bake for 14 minutes more, until tops of cupcakes spring back when touched and chocolate is not visible. Cool cupcakes in pan for 10 minutes. Remove from pan and cool completely on a wire rack.
4. For frosting, in a medium saucepan over low heat, bring 1 inch of water to simmering. Place chopped milk and semisweet chocolate in a medium mixing bowl. Place the bowl over the saucepan of water. Stir chocolate with a rubber spatula until melted. Remove from heat; cool chocolate for 15 minutes. With electric mixer on low speed, beat chocolate for 30 seconds. Beat in butter, 1 to 2 tablespoons at a time. Beat in powdered sugar until smooth. Spread frosting on cupcake tops.