In a large bowl, whisk together the sweetened condensed milk, cocoa powder, vodka, vanilla and peppermint extract. The mixture will be thick.
In a separate bowl, use an electric mixer (stand or hand-held) to whip the cream until soft peaks form.
Whisk 1/3 of the whipped cream into the sweetened condensed milk mixture to loosen it up.Fold another 1/3 of the whipped cream in until mostly combined.Add the last 1/3 of the whipped cream, as well as the chopped candy canes and chocolate. Fold everything together, leaving a few streaks of white.
Transfer the ice cream to lidded ice cream containers, and freeze until firm, about 6 hours.
Notes
Yields 1 1/2 quarts.
Keyword Chocolate, Christmas, Ice Cream, No Churn Ice Cream, Peppermint