Whisk the instant chocolate pudding mix with the cold whole milk for about 2 minutes, until thickened. Spread the pudding evenly into the prepared 9-inch chocolate cookie crumb crust.

Beat the softened cream cheese and powdered sugar until smooth and creamy. Mix in the peppermint extract, taking care to use peppermint extract rather than mint extract for the intended flavor.

Fold approximately half of the thawed whipped topping and the crushed peppermint candies into the cream-cheese mixture. Fold gently and stop as soon as the mixture is combined so the filling keeps its volume.

Spread the peppermint cream-cheese layer over the chocolate pudding layer. Add the remaining whipped topping over the surface and garnish with additional crushed peppermint candy, if desired.

Refrigerate the pie for at least 4 hours before serving; overnight chilling gives the cleanest slices. For especially neat cuts, freeze the refrigerated pie for 30 to 45 minutes, then slice with a warm, dry knife.