Line a 9-by-5-inch loaf pan with heavy-duty foil, leaving an overhang on all sides. Arrange 5 ice cream sandwiches snugly in the bottom of the pan. Spread the strawberry ice cream on top in an even layer. Press the remaining 5 ice cream sandwiches on top, then spread the mint chip ice cream evenly over the sandwiches. Top with an even layer of crushed cookies, pressing gently.
Cover the cake with plastic wrap and freeze until set, about 4 hours. Uncover, then lift the foil to remove the cake from the pan. Invert onto a platter, remove the foil and spread with the chocolate shell topping. Serve immediately or cover and freeze for up to 2 days.