Whisk the eggs with the sugar until foamy and white. Add the flour, cornstarch, and baking powder, folding everything into the egg and sugar mix.
Spread the cake mix on a baking tray and bake at 350°F for 10 minutes. Once cooled, cut the cake into 4 equally-sized pieces to form the base of the ice cream cake. Place these pieces inside the prepared milk carton.
Brush espresso onto the first layer of cake in the carton. Mix the vanilla ice cream and cream cheese, then spread it on top of the cake layer. Sprinkle half of the walnuts, add melted chocolate, and whipped cream.
Add another layer of cake, then repeat the process with espresso, ice cream, walnuts, melted chocolate, and cream.
Freeze the ice cream cake for three hours. Sprinkle cocoa powder on top before serving.