Step 1

Soften the butter. Cut 1 stick of unsalted butter into small cubes, then place in the bowl of a stand mixer. (Alternatively, place in a large bowl if using an electric hand mixer.) Let sit at room temperature until the butter is softened, about 30 minutes.

Step 2

Preheat the oven and prepare the baking pan. Arrange a rack in the middle of the oven and heat the oven to 350°F. Lightly coat a half sheet pan (18x13 inches) with cooking spray, then line with parchment paper so that it hangs off the short sides and forms a sling. Lightly coat the parchment paper with cooking spray.

Step 3

Combine the dry ingredients. Place 1 1/4 cups all-purpose flour, 1/2 cup unsweetened Dutch-process cocoa powder, 1/2 teaspoon baking soda, and 1/2 teaspoon kosher salt in a medium bowl and whisk to combine.

Step 4

Beat together the butter, sugars, egg, and vanilla. Add 1/2 cup packed light brown sugar and 1/3 cup granulated sugar to the bowl of butter. Beat on medium speed with the paddle attachment until light and fluffy, about 3 minutes. Scrape down the sides of the bowl. Add 1 large egg and 1 teaspoon vanilla extract and beat until combined, about 1 minute. Scrape down the sides and bottom of the bowl.

Step 5

Add the flour mixture and milk. With the mixer on low speed, beat in 1/2 of the flour mixture, then 1/3 cup whole milk. Scrape down the sides and bottom of the bowl. Beat in the remaining flour mixture until just combined.

Step 6

Spread the dough into the pan. Dollop the dough into the prepared pan. Coat an offset or rubber spatula lightly with cooking spray. Use it to spread the dough out, as if frosting a cake, into a thin, even layer that goes all the way to the edges of the pan.

Step 7

Bake the cookie. Bake until the edges are firm and have pulled away from the sides slightly, and a toothpick inserted into the center comes out mostly clean with just a few crumbs, 10 to 12 minutes.

Step 8

Chill the cookie. Place the pan on a wire rack and poke the cookie all over with a skewer or chopstick. Let cool for 5 minutes. Grasping the excess parchment, lift the cookie out of the pan and set it on the rack. Transfer the cookie on the rack to the freezer to chill. (Alternatively, if you don’t have space in the freezer, chill in the refrigerator). Place a quarter sheet pan or 9x13-inch baking pan in the refrigerator to chill. Meanwhile, soften the ice cream.

Step 9

Soften the ice cream. Let 1 quart vanilla ice cream sit at room temperature to soften until spreadable but still firm, 15 to 20 minutes.

Step 10

Cut the cookie. Remove the cookie from the freezer. Gently invert a large cutting board over the cookie, then flip both the board and sheet pan over in one motion. Peel off the parchment paper, then cut the upside-down cookie in half crosswise. Remove the chilled pan from the freezer, fold the used parchment paper in half, and place it in the baking pan. Place an upside-down cookie half in the pan.

Step 11

Assemble and freeze the ice cream sandwich slab. Use an ice cream scoop to spoon the softened ice cream all over the top, then spread into an even layer with an offset or rubber spatula. Press the remaining cookie half, right-side up, on top of the ice cream. Freeze until firm, at least 4 hours.

Step 12

Cut the ice cream sandwiches. To serve, transfer the ice cream sandwich slab to a cutting board and cut into 12 individual ice cream sandwiches. Serve immediately or wrap each sandwich well in parchment paper or plastic wrap and return to the freezer until ready to serve.