In a 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 saucepan.
Slowly beat the hot half-and-half into the egg mixture.
Pour the entire mixture back into the pan and place over low heat.
Stir constantly using a whisk or wooden spoon until the custard thickens slightly (be careful not to let the mixture boil or the eggs will scramble).
Remove from the heat and pour the hot custard through a strainer into a large, clean bowl.
Let the custard cool slightly, the stir in the cream, peppermint extract, and food coloring; cover and refrigerate until cold or overnight.
Stir the chilled custard, then freeze in 1 or 2 batches in an ice cream maker by following the manufacturer’s directions.
When finished, the ice cream will be soft but ready to eat; for firmer ice cream, transfer to a freezer-safe container and freeze several hours.
*After-Dinner Mint Ice Cream: add ¾ cup chopped chocolate-covered mint candies to the machine when the ice cream is semifrozen; allow the machine to mix in the candy; proceed as directed.
*Grasshopper Pie Ice Cream: add ½ cup white creme de cacao to the custard before freezing; proceed as directed, adding ½ cup crushed chocolate wafers to the machine when the ice cream is semifrozen; allow the machine to mix in the wafers.