In a medium saucepan over medium heat, combine cream, milk, and sugar and stir until the sugar dissolves and the mixture is hot, but not boiling.
In a small, heat-resistant bowl, whisk the egg yolks. Slowly pour half the heated milk mixture into the bowl with the egg yolks, whisking the entire time. Pour the egg yolk mixture back into the saucepan with the rest of the milk, continuing to whisk.
Cook mixture over medium heat until it begins to thicken and coats the back of a wooden spoon, about 5-7 minutes. Strain the mixture into a heat-resistant bowl.
Cover bowl with plastic wrap so that plastic is touching top of cream mixture (this will prevent a skin from forming). Refrigerate until chilled, at least 4 hours.
Once mixture is chilled, stir in peppermint extract and freeze in an ice cream maker according to the manufacturer’s instructions. When ice cream is almost frozen, mix in crushed candy canes and marshmallow fluff.
When ice cream is frozen, remove to airtight containers and place in freezer for at least 4 hours to continue to harden.