Add a few drops of peppermint extract to vanilla frosting and mix until incorporated. Use a knife to spread the frosting around the rims of 3 glasses, then roll the glasses on a plate covered with the white chocolate chips. Chill the glasses in the freezer while you make your milkshake.
Combine the milk and ice cream in a blender and blend on high. Pour half the milkshake into a pitcher and set aside. Add 2 to 4 drops red food coloring and 1 teaspoon peppermint extract to the milkshake remaining in the blender and blend until smooth. Starting with one color, pour a layer of milkshake into each prepared glass and freeze for 10 minutes. Pour in a layer of the other color and freeze for 10 minutes more. Continue layering until you have made several stripes. Top the milkshakes with whipped cream, white chocolate chips, crushed candy cane bits or sprinkles, and 1 or 2 whole candy canes.