Freeze the base of your ice cream maker overnight.
In a medium saucepan, whisk together 1 cup of cold whole milk and the cornstarch.
Add the granulated sugar and place the saucepan over medium heat. Whisk continuously for 4 to 6 minutes until the mixture thickens.
Remove from heat and stir in the remaining ½ cup of whole milk, heavy cream, salt, peppermint extract, and vanilla extract.
Transfer the mixture to a large bowl. Cool in an ice water bath for about an hour, or refrigerate for 3-4 hours until it reaches 40°F.
Churn the mixture in your ice cream maker according to the manufacturer's instructions, typically about 25 minutes.
In the last minute of churning, add the crushed peppermint candy.
Transfer the ice cream to a loaf pan or airtight container, sprinkle extra crushed candy on top if desired, cover, and freeze for at least 2 hours or until firm.
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Notes
Store in an airtight container in the freezer for up to 2 weeks.