In a medium bowl, combine the crushed chocolate cookies and the melted butter. Stir until the crumbs are evenly coated. Press the mixture firmly into the bottom of a 9 by 13 inch baking dish to create an even layer. Place the dish in the refrigerator to chill while you prepare the next layer.
In a large mixing bowl, beat the softened cream cheese until it is smooth and fluffy. Add the granulated sugar, 2 tablespoons of milk, alcohol-free vanilla extract, and peppermint extract. Mix until well combined. Gently fold in 1 and a half cups of the whipped topping. Spread this mixture carefully over the chilled cookie crust using a spatula.
In another large bowl, whisk together the two packages of instant chocolate pudding mix and the 3 1/4 cups of cold milk. Continue whisking for about two minutes until the pudding starts to thicken. Let it sit for a moment to set further, then spread the pudding evenly over the peppermint cream layer.
Spread the remaining whipped topping over the pudding layer until it is completely covered. Cover the dish with plastic wrap and refrigerate for at least 4 hours, or ideally overnight, to allow the layers to set properly and the flavors to meld together.
Just before serving, sprinkle the crushed candy canes and mini chocolate chips over the top. Using a sharp knife, cut the dessert into squares and serve cold.
