Special equipment: 12 serving glasses; a funnel
In the bowl of a stand mixer fitted with the paddle attachment, mix the ice cream, 1/4 cup of the crushed peppermint candy and the peppermint extract until smooth. Transfer to a large mixing bowl, and whisk in the milk and then the club soda to make the punch frothy.
Place 1 cup cold water in a small bowl, and place the remaining 3/4 cup crushed candies in another small bowl. Dip the rims of 12 glasses in the water and shake off any excess. Then dip the wet rims in the bowl of the crushed candies. Using a funnel to avoid touching the candied rims, pour 8 ounces of punch into each glass. Serve immediately.
You can use store-bought peppermint ice cream instead of flavoring your own. Store the punch for up to 2 hours in the refrigerator, whisk vigorously before serving.