1. PREHEAT oven to 350°F.
2. Line two 9” round cake pans with parchment paper rounds and spray with baking oil.
3. Whisk together all-purpose flour, baking powder, and salt in a medium bowl; set aside.
4. Remove chilled cookie dough from the refrigerator and turn out onto a cutting board. Knead and press into a 1-inch thick oval shape. Cut into squares (roughly ¾ inch each), and roll each into a little ball, about ¾-inch in diameter. Divide into two piles; set aside. (The balls can be shaped while the butter is creaming, if using a stand mixer.)
5. Cream butter and granulated sugar using a stand or electric mixer on high about 5 minutes, or until light and fluffy, scraping down sides of the bowl with a rubber scraper as needed. Reduce speed to medium and add eggs one at a time, mixing well after each addition. Beat in vanilla extract.
6. Reduce speed to low and slowly add flour mixture in three parts, alternating with milk. Start and end with the dry ingredients. Mix until just combined, but smooth.
7. Pour about ¼ of the cake batter into each of the two pans and smooth with a rubber scraper. Evenly divide the first half of the cookie dough balls between the two pans, placing the balls on the batter surface about 1 to 2 inches apart so that each slice will be embedded with dough.
8. Divide remaining cake batter equally between the two pans, making sure the cookie dough drops are covered and the batter is smooth. Divide the remaining pile of dough balls between the two pans again, pressing them into the batter ever so slightly.
9. Bake for 40 to 45 minutes, or until a cake tester inserted into the center of the cake comes out clean and cake springs back when lightly pressed in the center. The cake will be golden brown and may shrink away from the sides of the pan.
10. Cool cakes in the pans on a wire rack for 45 minutes. Run a thin-bladed knife or offset metal spatula around the outside of the cake to loosen from the pan. Carefully invert onto racks and remove pans to cool completely before assembly.