If you love cookie flavored ice creams, this ice cream sandwich will be the perfect treat. A mix of cookies and cream and cookie dough ice creams are sandwiched between fudgy cake that has a soft, chewy texture.
Preheat the oven to 375˚ F. Coat a 13-by-18-inch rimmed baking sheet with cooking spray and line with parchment paper. Spray the parchment paper with cooking spray. Heat the milk in the microwave until steaming, 30 to 45 seconds. Put the cocoa powder in a large bowl; pour the hot milk on top and whisk to combine. Let the cocoa steep, about 5 minutes.
Add the egg yolks, 1/2 cup sugar, the vegetable oil, vanilla and espresso powder to the cocoa mixture; whisk to combine. Sift in the flour, baking powder and salt; whisk to combine.
Combine the egg whites and cream of tartar in a separate large bowl. Beat with a mixer on medium speed until soft peaks form, 2 to 3 minutes. Increase the speed to medium high and beat in the remaining 1/4 cup sugar, 1 tablespoon at a time, until stiff peaks form, about 3 minutes. Whisk one-third of the egg white mixture into the batter to lighten. Gently fold in the remaining egg white mixture in 2 batches until fully combined and there are no white streaks remaining.
Scrape the batter onto the baking sheet; spread evenly to the edges. Bake until the cake just springs back to the touch and no longer looks wet in the center, 12 to 16 minutes. Transfer to a rack and let the cake cool 10 minutes in the pan, then invert onto the rack and carefully peel off the parchment. Let cool completely, about 30 minutes.
Transfer the cake to a cutting board, cut in half crosswise and lightly wrap each half in plastic wrap. Freeze the cake halves until almost frozen and firm, about 2 hours.
In the last 15 to 30 minutes of freezing the cakes, remove the ice cream from the freezer and let soften until spreadable but not melted. Line a 9-by-13-inch baking dish with plastic wrap, leaving a 6-inch overhang on the sides. Unwrap the cake halves; place 1 cake half in the bottom of the dish. Scoop the cookie dough ice cream all over the cake and spread into an even layer with an offset spatula, leaving a 1/8-inch border of cake. Top with the cookies-and-cream ice cream and spread into an even layer. Top with the remaining cake half and gently press down. Wrap tightly with the overhanging plastic wrap. Freeze the cake in the baking dish until the ice cream is firm (the cake will stay soft), at least 4 hours or overnight.
Lift the whole sandwich from the baking dish and remove the plastic wrap. Using a sharp knife, trim the edges, then cut into pieces.