Spray a 9x13-inch baking dish with baking spray or line it with parchment paper, leaving an overhang on the long sides for easy lifting.
Place the Oreo cookies in a food processor or a sealed zip-top bag and crush them into fine crumbs. Reserve 1 cup of the crumbs for the topping.
In a bowl, mix the remaining crumbs with the melted butter. Press this mixture evenly into the bottom of the prepared baking dish to form a firm crust.
In a large bowl, beat the softened cream cheese, vanilla, and powdered sugar on medium speed until very smooth and no lumps remain, about 1-2 minutes.
Gently fold in half of the whipped topping just until the mixture looks thick, light, and uniform, with no streaks of cream cheese or whipped topping.
Spread the cream cheese mixture evenly over the cooled Oreo crust, taking care not to disturb the crumbs.
In another medium bowl, whisk together the instant chocolate pudding mix and cold milk for about 2 minutes, until the mixture is smooth and thick enough that it lightly coats the back of a spoon.
Let the pudding stand for 2 to 3 minutes to finish thickening. It should be pourable but not runny.
Carefully pour or spoon the pudding over the cream cheese layer and spread it into an even layer, all the way to the edges.
Spread the remaining whipped topping evenly over the pudding layer. Sprinkle the reserved Oreo crumbs on top for added crunch.
Cover the dish and refrigerate for at least 4 hours, up to overnight, until the layers are fully set.
Once chilled, slice and serve cold.