In a large bowl beat the sugar into the eggs until thick and pale yellow. Add the cornstarch and beat until combined. Set aside.
Bring the half and half 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. Don't let it come to a boil, or the eggs will scramble.
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, then stir in the cream and peppermint extract. Cover with plastic wrap and refrigerate until very cold, at least 2 hours or overnight.
Stir the chilled custard, then make the ice cream according to your ice cream maker's manufacturers instructions. When ice cream is still soft, but coming together( 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.