Line an 8×8-inch pan with parchment paper, leaving an overhang for easy removal. Set aside for now.
Remove 1 heaping cup of whipped topping and set it aside as well.
In a large bowl, combine both pudding mixes and milk and briskly whisk together for about 1 minute or until smooth.
Take the remaining whipped topping and gently fold it into the pudding mixture until completely incorporated.
Place down an even layer of cookies in the prepared pan, breaking them apart as needed to fit the pan.
Top with half the pudding mixture and use the back of a large spoon to gently spread into an even layer.
Top with one more layer of cookies.
Add the rest of the pudding, again spreading it into an even layer.
Add the last layer of cookies.
Finally, spread on the reserved cup of whipped topping in an even layer.
Cover the cake and refrigerate for 8 hours.
When ready to serve, cut the cake into squares and, if desired, crumble a few reserved cookies on top as a garnish.