Preheat oven to 350 degrees F. Line a cupcake pan with paper liners.
In a mixing bowl, whisk together flour, cocoa powder, sugars, baking soda, baking powder, and salt. Make a well in the center of the dry ingredients and add milk, oil, vanilla, and eggs. Stir until just combined, scraping bottom of bowl with a rubber spatula to incorporate all of the flour.
Fill cupcake liners with scant 1/4 cup of the batter (cups should be just less than two-thirds full). Bake 18 to 22 minutes, or until a toothpick inserted in the center comes out clean. Transfer to a cooling rack to cool completely.
While cupcakes cool, prepare the cookie dough. In a large mixing bowl, beat together butter and sugars with an electric mixer on medium speed until light and fluffy, 2 to 3 minutes. Beat in milk and vanilla. Add flour and salt and mix on low speed (or by hand) until incorporated. Set aside 1 cup of the dough for the buttercream; stir 1/3 cup of the chocolate chips into the remaining dough.
To prepare the buttercream, cream butter with an electric mixer until fluffy, 1 to 2 minutes. Slowly add powdered sugar, 1/2 cup at a time, and beat until smooth. Add the reserved plain cookie dough and mix on medium-high speed until light and fluffy, about 2 to 3 minutes.
Using a paring knife, cut a cone-shaped piece (approximately 1 inch in diameter) from the top of each cooled cupcake; pinch off and dispose of (i.e., eat!) the tip of each cone. Fill cupcakes with remaining cookie dough and replace tops. Generously spread or pipe on frosting, being sure to cover the seams. Sprinkle with the remaining cup chocolate chips.
Reprinted with permission from The Cookie Dough Lover's Cookbook, by Lindsay Landis, copyright (c) 2012. Published by Quirk Books.