1. Place a rack in the center of the oven and preheat the oven to 350 degrees F. Line 24 cupcake cups with paper liners. Set the pans aside.

2. Place the cake mix, pudding mix, milk, oil, eggs, and vanilla in a large mixing bowl. Blend with an electric mixer on low speed for 30 seconds. Stop the machine and scrape down the sides of the bowl with a rubber spatula. Increase the mixer speed to medium and beat 2 minutes more, scraping down the sides again if needed. The batter should look well blended. Spoon or scoop a heaping 1/4 cup batter into each lined cupcake cup, filling it two thirds of the way full. (You will get between 22 and 24 cupcakes; remove the empty liners, if any.) Cut the frozen dough pieces in half to make 24 pieces. Place a frozen cookie dough piece on top of each cupcake. Place the pans in the oven.

3. Bake the cupcakes until they are golden and spring back when lightly pressed with your finger, 23 to 27 minutes. Remove the pans from the oven and place them on wire racks to cool for 5 minutes. Run a dinner knife around the edges of the cupcake liners, lift the cupcakes up from the bottoms of the cups using the end of the knife, and pick them out of the cups carefully with your fingertips. Place them on a wire rack to cool for 15 minutes before frosting. They may sink a bit in the center.

4. Meanwhile, prepare the Chocolate Buttercream.

- Place the butter and cocoa powder in a large mixing bowl. Blend with an electric mixer on low speed until the mixture is soft and well combined, 30 seconds. Stop the machine and add the confectioners' sugar, 3 tablespoons of the milk, and the vanilla. Blend with the mixer on low speed until the sugar is incorporated, 1 minute. Increase the speed to medium and beat until light and fluffy, 1 minute more. Add 1 to 2 tablespoons more milk if the frosting is too stiff.

5. Place a heaping tablespoon of frosting on each cupcake and swirl to spread it out with a short metal spatula or a spoon, taking care to cover the tops completely. The cupcakes are ready to serve.

Store these cupcakes, in a cake saver or under a glass dome, at room temperature for up to 3 days or in the refrigerator for up to 1 week. Or freeze them, wrapped in aluminum foil or in a cake saver, for up to 6 months. Thaw the cupcakes overnight in the refrigerator before serving.