Whisk French vanilla pudding mix with milk in a large bowl until softly set, about 2 minutes. Pudding will thicken as it stands. Cream 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 frozen whipped topping into the mixture.

Place half the cookie crumbs into the bottom of an 9x12-inch dish; retain remaining crumbs for topping. Spread the filling over the layer of cookie crumbs and top with remaining crumbs, covering completely. Chill for at least 30 minutes.