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 just to a boil over medium heat, stirring occasionally.
Slowly add half of the hot half-and-half mixture to the egg mixture, whisking constantly as you pour the hot half-and-half mixture in; then pour this mixture back into the saucepan.
Place the saucepan back over medium-low heat, and cook the custard, stirring constantly, until it thickens enough to coat the back of a spoon; keep a close eye on it, as it should never come to a boil. When thick enough, the custard should register 170°F on a candy thermometer. Pour the custard through a fine-mesh strainer into a large bowl.
Allow the custard to cool slightly, then stir in the cream and peppermint extract. Cover and refrigerate until cold or overnight.
Stir the chilled custard. Freeze the custard in an ice cream maker according to the manufacturer’s instructions.
Transfer the finished ice cream to an airtight container, layering and mixing in the crushed peppermint candies, and place it in the freezer until ready to serve.
When finished, the ice cream will be soft but ready to eat. For firmer ice cream, wait at least 2 hours, or overnight.
Notes
This recipe makes about 1 quart of ice cream. Serving size of 1/2 cup means about 8 servings.