Whip half a cup of heavy cream with the powdered sugar until soft peaks form. Keep chilled until ready to use.
In a small saucepan combine milk, remaining heavy cream, and sugar. Whisk over medium heat until mixture comes to a simmer. Remove from heat and add chocolate. Whisk until chocolate has melted and mixture is smooth. Stir in peppermint extract.
Divide among serving mugs and top with whipped cream and a candy cane.