In a medium mixing bowl, beat the sugar into the eggs until thickened and pale yellow. Beat in the cornstarch and set aside.
Bring the half-and-half to a simmer in a heavy medium saucepan. Slowly beat the hot half-and-half into the eggs and sugar. Pour the entire mixture back into the pan and place over low heat. Stir constantly with a wooden spoon until the custard thickens and coats the back of the wooden spoon. Be careful not to let the mixture boil or the eggs will scramble. Remove from heat and pour the hot custard through a strainer into a large clean bowl. Allow the custard to cool slightly, and stir in the cream, peppermint extract, and food coloring, if using. Cover and refrigerate until cold or overnight.
Stir the chilled custard, and freeze in your ice cream machine according to the manufacturer’s directions. Slowly add any mix-ins to the machine when the ice cream is semifrozen, or in the last couple minutes of mixing. Allow the machine to mix in the chocolate. When finished, the ice cream will be soft but ready to eat. For firmer ice cream, transfer to a freezer safe container and freeze at least 2 hours.
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