Combine the dairy and sugar. In a large mixing bowl, blend the whole milk, light cream, heavy cream, and granulated sugar. Stir thoroughly until the sugar has completely dissolved into the dairy mixture.
Add the extracts and salt. Pour in the vanilla extract and peppermint extract, then add the salt. Stir well so the flavorings are evenly distributed throughout the mixture.
Incorporate the candy canes. Fold in the crushed candy canes, reserving a small amount for garnishing the finished ice cream. As the candy pieces begin dissolving, they will naturally tint the mixture pink while intensifying its peppermint flavor.
For more noticeable pieces of peppermint candy in the finished ice cream, do not add the crushed candy canes at this stage. Instead, incorporate them during the final 5 minutes of churning.
Churn the mixture. Transfer the prepared ice cream base to an ice cream maker. Churn according to the manufacturer’s instructions until the mixture develops a soft-serve consistency.
Serve or freeze. Serve the peppermint ice cream immediately for a softer texture, or transfer it to a suitable freezer container and freeze until it reaches a firmer consistency.
Before serving, scatter the reserved crushed candy canes over the top for a festive finishing touch.