Dissolve Candy Cane. To start, pour the whole milk into a large liquid measuring cup or a small bowl. Add 1/2 cup of the crushed candy cane pieces and stir to dissolve until the milk turns a pretty pink color and there are no large pieces of candy cane left. (This can take up to 10 or so minutes!)
Combine. Then, add sugar, vanilla extract, peppermint extract, and salt. Stir to combine until the sugar is dissolved. You don't want any gritty grains left! Pour in the heavy cream and mix through until evenly blended. If your mixture has warmed up to room temperature, make sure to chill until it feels cold to the touch, or it won't churn properly. (I usually don't need to chill mine!)
Churn Ice Cream. Pour the ice cream base into the ice cream maker and churn according to your manufacturer's instructions until the ice cream reaches a soft serve texture. (For me, that's 20-25 minutes!) Sprinkle in most of the remaining 1/2 cup crushed candy cane (I like to reserve about 1 Tablespoon!)
Freeze. Transfer ice cream mixture to a freezer-safe container and sprinkle with any remaining candy cane. Freeze at LEAST 6-8 hours on a level surface in the freezer, or until it's firm enough to scoop.
Store. Leftover ice cream will keep in an airtight container in the freezer up to 2 weeks (though we prefer to enjoy it within 2-3 days, for the best texture!)