01 - Combine cream, milk, and half of sugar in a medium saucepan. Heat over medium until just steaming, not boiling.
02 - Whisk egg yolks with remaining sugar in a mixing bowl until pale and slightly thickened.
03 - Gradually pour about 1 cup of hot milk mixture into egg yolks while whisking constantly to temper.
04 - Pour tempered yolks back into saucepan. Cook over low heat, stirring constantly with a wooden spoon until custard thickens and coats the back of the spoon (170-175°F).
05 - Remove from heat. Stir in peppermint extract, vanilla extract, and green food coloring until well combined.
06 - Strain custard through a fine mesh sieve into a clean bowl. Cool to room temperature, cover, and refrigerate for at least 4 hours or until thoroughly chilled.
07 - Churn chilled mixture in an ice cream maker according to manufacturer's instructions. Add chopped chocolate chips during the last few minutes of churning.
08 - Transfer ice cream to a freezer-safe container. Freeze for at least 2 hours or until firm.
09 - Scoop and serve immediately.