Special equipment: an 18-by-13-inch rimmed baking sheet
Preheat the oven to 375 degrees F. Line an 18-by-13-inch rimmed baking sheet with foil, leaving a 2-inch overhang on at least two sides. Coat with cooking spray.
Arrange the chocolate sandwich cookies on half of the baking sheet in 6 rows, with 4 cookies per row. Cover the chocolate sandwich cookies with the cookie dough: pressing and spreading the dough into and over the cookies to make one giant chocolate chip cookie that covers half of the baking sheet.
Prepare the brownie batter according to package directions. Pour the batter into the empty half of the pan. Arrange the vanilla sandwich cookies on top of the brownie batter in 6 rows, with 4 cookies per row Press the cookies into the batter.
Bake until a toothpick inserted in the center of the chocolate chip cookie portion and brownie portion comes out with just a few crumbs, 20 to 23 minutes. Let cool completely in the pan on a cooling rack, about 2 hours. Freeze to firm up, about 30 minutes.
Use the foil overhang as handles and lift the giant cookie out of the baking sheet and carefully flip it over onto a cutting board, making sure it does not bend or crack. Peel away the foil from the bottom. Line the baking sheet with plastic wrap.
Using a warm serrated knife, trim the rough edges of the giant cookie so it is an even rectangle (about 12-by-16-inches). Cut it in half crosswise so you have one brownie half and one chocolate chip cookie half (each 8-by-12-inches). Lay the chocolate chip cookie piece on the prepared baking sheet so the chocolate sandwich cookies are facing up. Working quickly, spread the ice cream across the cookie all the way to the edges. Smooth out the top and sides. Place the brownie half, with the chocolate cookies facing down, on top of the ice cream and gently press to adhere. Fold the plastic wrap around the assembled ice cream sandwich to help hold its shape and freeze until hardened, at least 6 hours and up to overnight. Use a warm serrated knife to cut into 24 ice cream sandwiches.
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