In a saucepan, combine milk, heavy cream, ½ cup sugar, and salt. Heat over medium until steaming, not boiling.
In a bowl, whisk egg yolks with remaining ¼ cup sugar until slightly thickened.
Slowly pour the warm cream mixture into the yolks while whisking constantly.
Return the mixture to the saucepan. Cook over medium-low heat, stirring constantly, until it reaches 170–175°F (77–80°C) or coats the back of a spoon.
Strain into a clean bowl. Stir in vanilla extract, peppermint extract, and food colouring (if using).
Cover and refrigerate for at least 4 hours or overnight, until completely cold.
Churn in an ice cream maker until it reaches a soft-serve consistency (about 20–25 minutes).
Add chopped chocolate during the last few minutes of churning.
Transfer to a container and freeze for 4–6 hours, until firm and scoopable.