Make the ice cream base: Prepare an ice bath. Pour the whipping cream into a smaller bowl set in the ice bath; set aside. Combine the sugar and milk in a medium saucepan over medium heat, stirring until the sugar dissolves. Beat the egg yolks with a fork in a glass measuring cup. Gradually whisk about 1/2 cup of the hot milk into the yolks to temper, then pour the yolk mixture back into the saucepan. Cook, stirring, until it thickens a bit and coats the back of a spoon. Slowly pour the custard into the cream and stir in the vanilla. Let the bowl sit in the ice bath, stirring often, until cold (or chill in the fridge up to 24 hours). Process in your ice cream maker per the machine's instructions.
Make the fudge swirl: Combine the heavy cream and corn syrup in a saucepan over medium heat and cook until bubbling. Remove from heat and whisk in the chocolate and 1/2 tsp peppermint extract; taste and add a few more drops if desired. Cool to room temperature.
Assemble: Line a loaf pan with parchment. Using a big spoon, drop alternating dollops of churned ice cream and fudge, sprinkling in the crushed candy canes as you go. Use a knife to gently swirl the ice cream and fudge together a bit. Cover with plastic wrap and freeze until firm, about 4 hours. Remove from the freezer 10–15 minutes before serving — it should be on the soft side.
Notes
Temper the yolks: Whisk a little hot milk into the yolks first so they don't scramble when returned to the pan.Chill fully before churning: A cold base churns faster and freezes smoother — you can refrigerate it up to 24 hours.Swirl gently: Fold the fudge and candy canes in with just a few passes of a knife; over-swirling muddies the marble.Soften before serving: Let it sit 10–15 minutes out of the freezer so it scoops soft.