Whisk milk and pudding mix in a large bowl until softly set, about 2 minutes. The pudding will thicken as it stands.
Beat confectioners' sugar, cream cheese, and butter in a separate bowl until smooth and creamy. Stir cream cheese mixture into pudding until thoroughly combined; gently fold in frozen whipped topping.
Place ½ of the cookie crumbs into the bottom of a 9x12-inch dish; spread evenly. Cover cookie crumbs with pudding mixture and top with remaining crumbs, covering completely.
Chill for at least 30 minutes before serving.
