Line a 9x13-inch baking dish with parchment paper or spray with nonstick cooking spray.
Use a food processor to crush the candy canes until coarsely crushed and ¾ cup in volume. Set aside.
Use a food processor to turn the Oreos into fine crumbs. Add the melted butter and mix.
Press the mixture into the baking dish. Store in the freezer while you prepare the next layer.
In a large bowl, beat the cream cheese with an electric mixer.
Add the white sugar, powdered sugar, and Cool Whip to the cream cheese and mix until combined. Make sure not to over-mix the cream cheese layer.
Stir ½ cup of the crushed candy canes into the cream cheese mixture. Set aside the remaining ¼ cup for the topping.
Use a spatula to smooth this layer over the prepared crust. Put the baking dish back into the freezer while you prepare the next layer.
Combine the dry pudding mix and milk and beat for two minutes until it starts to thicken.
Use a spatula to evenly smooth the pudding layer over top of the cream cheese layer. Put it back into the freezer for ten minutes.
Smooth the remaining Cool Whip over the top of the dessert and sprinkle with the remaining crushed candy cane pieces.
Place the dish into the refrigerator for at least six hours before slicing and serving. For the best layers, it should chill overnight.