If you're using a type of ice cream maker that requires a frozen container, place it in the coldest part of your freezer and freeze for 12-24 hours.
In a medium saucepan, heat the half and half and cream over medium heat until just scalded. Remove from heat.
Add the sugar and eggs into a large bowl. Use a hand mixer to beat the egg mixture together until pale yellow and thick, about 3-4 minutes. Add the cornstarch and mix to combine.
In a slow, steady stream, add the scalded cream to the egg mixture, beating constantly.
Pour the cream mixture back into the saucepan and heat over medium to medium high heat, stirring and scraping the bottom of the pan constantly, until custard thickens. Remove from heat.
Set a fine mesh sieve over a large bowl and pour the custard through the sieve to collect any egg pieces.
Stir in the peppermint extract. Cover the custard with a piece of plastic wrap and press the plastic directly onto the surface of the custard to prevent a "skin" from forming. Refrigerate 2 hours or overnight. Custard must be very cold before starting the ice cream.
Churn the ice cream in your ice cream maker according to the manufacturer's instructions. Toward the end of the churning, add crushed peppermint candies and continue to churn until it's at the stage of firm soft-serve.
Layer the peppermint stick ice cream and peppermint magic shell in a container, alternating between the ice cream and magic shell. Finish with a drizzle of magic shell and a sprinkle of crushed peppermint sticks.