In a bowl, stir the cream and sugar together until the sugar dissolves. Add the vanilla and stir to combine.
Set aside for about 15 minutes, stirring occasionally, to allow the flavors to blend.
Cover and refrigerate until chilled, at least 3 hours or up to 8 hours.
In a food processor, grind the peppermint candies by pulsing 2 or 3 times. Some of the candies will be more finely ground than others; small chunks of candy will add texture and interest to the final ice cream.
Pour the cream mixture into an ice cream maker and freeze according to the manufacturer's instructions.
When nearly frozen and the consistency of thick whipped cream, add the ground peppermint candies. Churn or stir just until mixed.
Transfer the ice cream to a freezer-safe container.
Cover and freeze until firm, at least 3 hours or up to 3 days, before serving.