In a medium saucepan over medium heat, warm the cream, milk, half of the sugar (1/4 cup) and salt.
In a medium bowl, whisk together the yolks to break them up, and then whisk in the remaining ¼ cup sugar. Scoop out about ½ cup of the warm cream mixture from the saucepan and, while whisking the eggs constantly, add the cream mixture to the bowl with the egg yolks. Repeat, adding another ½ cup of the hot cream mixture to the bowl with the yolks. Using a heatproof spatula, stir the cream in the saucepan as you slowly pour the egg and cream mixture from the bowl into the saucepan.
Cook the mixture, stirring constantly with the spatula, until the mixture thickens and coats the back of the spatula. Strain the mixture through a fine-mesh strainer into a clean container. Set the container into an ice bath until cool. Chill the mixture thoroughly in the refrigerator.
Once the ice cream mixture is chilled, whisk in the peppermint extract and then freeze the mixture in your ice cream maker according to the manufacturer’s instructions. Fold the chocolate chips into the ice cream as you remove it from the ice cream maker.