Warm 1 cup of the heavy cream, half and half, granulated sugar and salt in a medium saucepan over medium-low heat, stirring occasionally, until the sugar dissolves.
In a large bowl, whisk together the egg yolks. Slowly pour the warm mixture into the egg yolks, whisking constantly, then pour mixture back into the saucepan.
Stir the mixture constantly over medium heat with a heatproof spatula, scraping the bottom as you stir, until the mixture thickens and coats the spatula (if using a thermometer, it should be right around 170°F).
Pour the remaining 1 cup of heavy cream into the large bowl and set a mesh strainer on top. Pour the custard through the strainer and stir it into the cream.
Add the vanilla and peppermint extracts and the food coloring (if using), and stir until cool over an ice bath. Chill the mixture thoroughly in the refrigerator (preferably overnight).
Churn according to your ice cream maker’s directions, adding the chopped chocolate during the last 5 minutes of churning.