In a medium pot, bring cream, milk, and 1 Tbsp of sugar to boil. Meanwhile, in a large bowl whisk together egg yolks and remaining sugar. Once milk begins to boil, lower heat.
Temper hot milk in sugar-yolk mixture. Slowly add hot milk into yolks about ¼ cup at a time, whisking to distribute heat. Once all the milk has been added, transfer mixture back into the pot.
Stir over low-medium heat until mixture reaches nappe consistency, or 180°F on a kitchen thermometer. Remove from heat and stir in vanilla. Transfer to an airtight container and let mixture mature in fridge overnight (at least 6 hours).
Freeze mixture in ice cream maker according to your unit's instructions. Once the volume has increased by half and has reached a soft serve consistency, add crushed candy cane. Continue to churn until candy cane has been distributed. Transfer ice cream to a freezer safe container. Freezer for at least 30 minutes before serving.