Using a food processor, pulse Oreos until finely crushed. Melt unsalted butter in the microwave, then combine with the crumb mixture until well blended. Firmly press this mixture into a 9x13-inch dish for a solid base. Place the crust in the refrigerator to chill for at least 10-15 minutes.

In a mixing bowl, beat brisk style cream cheese, powdered sugar, vanilla extract, and peppermint extract until smooth and creamy. Gently fold in stabilized whipped topping and some crushed candy canes. Carefully spread the cheesecake mixture over the chilled Oreo crust.

Return the dessert to the refrigerator for about 10-15 minutes to firm up the peppermint cheesecake.

In a separate bowl, whisk together the instant chocolate pudding mix and milk until thickened, about 2 minutes. Pour this pudding over the set cheesecake layer and smooth it evenly.

Allow the pudding layer to set in the refrigerator until firm, about 5-10 minutes. Spread a generous layer of stabilized whipped topping over the chocolate pudding.

Just before serving, sprinkle crushed candy canes across the top for a festive touch. Chill the dessert for at least 5 hours or overnight for best flavor.