Mix the sugar, cornstarch and eggs together in a bowl and set it next to the stove.
In a heavy saucepan set over medium heat, bring the 1 cup of milk and 1 cup of the cream to a simmer. Pour the simmering cream mixture into the egg mixture whisking constantly, then return the mixture to the saucepan. Reduce heat to about medium low and whisk or stir constantly until the mixture thickens – it should not come to a complete boil, but rather thicken just to the consistency of thick gravy and coat the spoon. If you have a candy thermometer, it should read somewhere between 165 and 170. Remove from heat and pour into a large bowl or a large shallow dish (a trick to make it cool faster).
Pour the remaining 1 cup of cream into the mixture, then stir in the peppermint extract and vanilla. If desired, add a few drops of green food coloring.
Cover loosely with plastic wrap and chill for a few hours or overnight until very cold.
Following the ice cream maker’s directions, mix the ice cream until thick. During last 5 minutes of ice cream mixing, add the grated chocolate. Spoon into a lidded container and let sit in the freezer to ripen until you are ready to serve it.
Makes about 4 cups