Heat oven to 350*F. Grease 24 muffin pan molds or line with paper liners.
Combine the cake mix, pudding mix, milk, oil, eggs, and vanilla in a large bowl; blend with an electric mixer on low speed for 30 seconds. Scrape down sides of bowl.
Increase mixer speed to medium and blend another 2 minutes, scraping down bowl as needed.
Scoop a heaping 1/4 cup batter into each cupcake well, filling at least 2/3 full. (you should get between 22 and 24 cupcakes; remove any empty paper liners).
Cut the frozen cookie dough pieces in half to make 24 pieces (or enough for how many cupcakes you wind up with); place a dough chunk on top of each cupcake.
Bake until cakes are golden and spring back when lightly touched, 23-37 minutes.
Cool on wire racks for 5 minutes, then carefully remove cupcakes from pans and cool another 15 minutes before frosting (they may sink a bit in the center). Serve warm.
Store cupcakes in a cake server at room temperature for up to 3 days or up to 1 week in the refrigerator. Bring to room temperature and then microwave for about 8 seconds to "re-goo" the texture.