Step 1

Line a 9-by-13-inch pan with two layers of plastic wrap, leaving a few inches of overhang on all sides. Open all of the ice cream containers. Place one 1 1/2 quart (or 3 pints) on the counter and let soften until it resembles thick cake batter, 5 to 10 minutes; you’ll use this ice cream first. Place the other containers in the refrigerator to soften while you build the first half of the cake. (Ice cream will reach the desired consistency in the refrigerator in 20 to 30 minutes.)

Step 2

Scatter the crushed cookies in a thin, even layer across the bottom of the pan. Spoon the first softened container of ice cream over the cookies and spread into an even layer. Press down as you go to adhere the ice cream to the cookies. If at any point the ice cream is melting to liquid, pop the cake and ice cream in the freezer to firm to a spoonable consistency.

Step 3

Place the pan in the freezer while you unwrap the sandwiches. Top the ice cream with a layer of ice cream sandwiches, cutting some as needed to fill the layer to the edges.

Step 4

Spoon the remaining ice cream over the top and spread into an even layer. Cover with plastic wrap and freeze until firm, at least 2 hours or up to 1 week.

Step 5

To serve, use the plastic overhang to remove the cake from the pan. Cut into squares or rectangles and serve with a drizzle of hot fudge or Magic Shell. If you’re finding the cake too solid to cut, don’t wait for it to soften: Warm your knife by running it under hot water and wiping it dry between cuts.