Line the bottom of a 9x13-inch pan with 12 ice cream sandwiches.
Open the jar of peanut butter, place into microwave oven, and microwave on low power for about 30 seconds at a time, until the peanut butter is warm and spreads easily. Spread half the peanut butter over the layer of ice cream sandwiches.
Mix the instant chocolate pudding mix into the whipped topping; spread half the mixture over the peanut butter layer.
Spread half the peanut butter cups over the whipped topping layer.
Remove tops of the hot fudge and caramel topping jars, and microwave on low power for about 10 seconds at a time just until the toppings are warmed and liquid. Drizzle half of the fudge and caramel sauces over the peanut butter cups.
Repeat the layers, starting with the remaining ice cream sandwiches, followed by remaining peanut butter, whipped topping mix, peanut butter cups, hot fudge sauce, and caramel sauce.
Freeze the dessert until solidly frozen, 1 to 2 hours.