01 - Heat the heavy cream and milk in a medium saucepan over medium heat until just steaming, taking care not to boil.
02 - Whisk together egg yolks, sugar, and salt in a separate bowl until pale and slightly thickened.
03 - Slowly pour approximately 1 cup of the hot cream mixture into the yolks while whisking constantly to temper the eggs.
04 - Pour the tempered yolk mixture back into the saucepan with the remaining cream.
05 - Cook over low heat, stirring constantly with a wooden spoon, until the mixture thickens and coats the back of the spoon (170-175°F), about 5-7 minutes.
06 - Remove from heat and stir in peppermint extract, vanilla extract, and food coloring if using.
07 - Pour the custard through a fine mesh sieve into a clean bowl. Cool to room temperature, then cover and refrigerate for at least 4 hours or until thoroughly chilled.
08 - Churn the chilled custard in an ice cream maker according to the manufacturer's instructions.
09 - During the final 2 minutes of churning, add the chopped chocolate so it disperses evenly throughout.
10 - Transfer the ice cream to a lidded container and freeze for at least 2 hours until firm before serving.