Pulse the Oreos in a food processor or crush them inside a Ziploc bag until they result in fine crumbs. Reserve ⅓ of the crumbles for later.
Pour the cookie crumbles into the bottom of a 9x13-inch baking dish or a trifle bowl and press them together firmly. If you want multiple layers place only half of the Oreo crumbs on the bottom.
In a large bowl with a handheld mixer, or using the paddle attachment of an electric mixer, beat the butter and cream cheese together until smooth. Sift powdered sugar into the bowl and mix it in before adding the vanilla and combining.
Using a rubber spatula, gently fold the Cool Whip into the mixture.
In a separate bowl, whisk together the vanilla pudding mixture and milk until smooth. Let the mixture sit for about 4 minutes until the pudding is thickened.
Fold the pudding into the cream cheese mixture and spread it evenly over the Oreo crumb crust.
Sprinkle the top of the cake with the remaining Oreo crumbs, cover the dish, and let it set in the refrigerator for about 4 hours before serving.