Preheat the oven to 350 degrees F. Line a 9-by-13-inch pan with foil and coat with cooking spray. Prepare the brownie mix; pour into the pan and bake as directed. Let cool, then remove from the pan.
Cut out three brownie circles: one the size of the bottom of the ice cream carton (3 inches), one the size of the middle of the carton (3 3/4 inches) and one for the top (4 1/2 inches).
Carefully peel off the ice cream carton and remove the ice cream in one piece. Cut the ice cream in half crosswise.
Assemble the cake upside down on a baking sheet as shown: Put the largest brownie on the bottom, then the wider half of the ice cream, then the medium brownie, then the remaining ice cream, then the smallest brownie. Freeze 1 hour.
Roll out the fondant into an 8-by-15-inch rectangle on a surface dusted with confectioners' sugar. Trim one long side to make it straight, then roll down the edge by about 1/8 inch to create a lip.
Whisk 1 cup heavy cream in a bowl until medium peaks form; spread on the top and sides of the cake. Wrap with the fondant, positioning the lip against the bottom edge of the cake. Trim the excess; set aside. Return to the freezer.
Knead the cocoa powder into the reserved fondant until evenly brown. Roll out into a 2 1/2-by-13-inch strip. Trim the long sides to make them straight, then use a chopstick to make indentations in the fondant, 1/4 to 1/2 inch apart.
Whisk the remaining 1/2 cup heavy cream until very soft peaks form. Remove a few tablespoons to a zip-top bag and snip a small corner. Dissolve the instant espresso powder in 2 teaspoons hot water, then stir into the remaining whipped cream.
Turn the cake right-side up. Use some water to attach the brown strip to the middle. Spoon the coffee whipped cream on top of the cake, then pipe zigzags of the plain whipped cream on top and drag the chopstick through it. Freeze until ready to serve.