Put a 9-by-5-by-3-inch metal loaf pan in the freezer to chill. Put the chocolate-peppermint candies and 3/4 cup of the heavy cream in a medium saucepan over medium heat and cook, stirring frequently, until the candies are melted, about 5 minutes. Cool completely.
Meanwhile, whip the remaining 2 cups heavy cream in a large bowl with an electric mixer on medium high until firm peaks form, about 2 minutes. Pour the condensed milk into a medium bowl and fold in 1 cup of the whipped cream with a rubber spatula until combined, then fold the lightened mixture back into the whipped cream until well blended. Gently fold in the marshmallows.
Remove 2 tablespoons of the chocolate-peppermint candy sauce and set aside. Pour a third of the ice cream mixture into the chilled pan, then top with half of the remaining chocolate-peppermint candy sauce, add a third more of the ice cream mixture, then the remaining chocolate-peppermint candy sauce. Cover with the remaining ice cream mixture and sprinkle the crushed peppermint puffs on top. Drizzle the top with the reserved chocolate-peppermint candy sauce. Loosely cover with plastic wrap and freeze until solid and scoopable, about 3 hours and up to overnight. Serve in a cup with a candy cane if using.
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