In the chilled bowl of a stand mixer fitted with a whisk attachment, whip the cream until stiff peaks form. Transfer whipped cream to a separate bowl, then return bowl to mixer.

In the mixing bowl, stir the sweetened condensed milk, vanilla, and peppermint extract together until combined. Add about 1/4 of the whipped cream into the condensed milk mixture, stirring until lightened. Pour the rest of the whipped cream into the mixing bowl and fold together until no more white streaks remain.

Fold in the chopped semi-sweet, white chocolate, and crushed candy canes, then pour custard into a freezer-safe container. Cover and chill for at least 6 hours, preferably overnight.