Place Oreo cookies, filling and all, into a food processor and pulse until they are fine crumbs. This can also be done with a large ziploc bag and a rolling pin instead. Add chopped candy cane Kisses and melted butter. Pulse again until the crumbs are evenly moistened.
Press the crust mixture into a 9x13-inch baking dish and set in the freezer for 10 minutes while preparing the next layer.
Beat the cream cheese and sugar in a large bowl with an electric hand mixer until creamy and smooth.
Add milk, vanilla extract, and peppermint extract. Beat again until combined.
Beat the heavy cream in a separate bowl until thickened and medium peaks start to form. Add half of the whipped cream to the cream cheese mixture along with the chopped candy cane kisses and fold together. Add remaining whipped cream and fold until evenly combined. The color from the Kisses should turn this layer a pink shade, but you could add a few drops of food coloring for even more color, if desired (I didn't).
Spread the cheesecake layer evenly over the set Oreo crust, then refrigerate or freeze while preparing the next layer.
Combine both packages of instant chocolate pudding mix in a large bowl with the cold milk. Whisk or beat with an electric mixer for 1-2 minutes until thickened.
Carefully pour the chocolate layer over the peppermint layer and spread to the edges of the pan. Transfer to the fridge to set up for at least 2-3 hours.
Shortly before serving, make the whipped topping by beating the remaining heavy cream, powdered sugar, and vanilla extract in a bowl until medium peaks form. Carefully spread the whipped cream over the set chocolate layer and sprinkle with the remaining chopped Candy Cane Kisses or crushed candy cane pieces. Refrigerate until ready to serve.