In a large bowl beat the Granulated Sugar (3/4 cup) into the large Egg (2) until thick and pale yellow. Add the Corn Starch (1 tablespoon) and beat until combined. Set aside. Bring the Half and Half (2 cup) to a simmer in a heavy medium saucepan.
While the mixer is running, very slowly stream the hot cream into the egg mixture. Pour the egg mixture back into the saucepan and heat over medium heat, stirring constantly until mixture thickens.
Set a fine mesh strainer over a large bowl and strain the custard through the sieve to remove any egg solids. Allow the custard to cool for 10 minutes.
Stir in the Heavy Cream (1 cup) and Peppermint Extract (2 teaspoon). Cover with plastic wrap and refrigerate until very cold, at least 2 hours or overnight.
Break the Chocolate Wafer Cookies (1/2 cup) into half-inch pieces.
Chop the premium Dark Chocolate Chips (1/2 cup) into smaller pieces.
Stir the chilled custard. Make the ice cream according to your ice cream maker's instructions. About three quarters of the way through the churning process, add the chocolate chips and cookie pieces.
When finished, transfer the ice cream to a freezer container and chill until firm, about 2 hours. Serve.