Warm the dairy: In a medium saucepan, add the milk and cream. Set over medium-low heat and warm gently until steaming, not boiling.
Whisk in dry ingredients: Add the cocoa powder, sugar, and salt.
Whisk until the cocoa is fully dissolved and the mixture looks smooth and chocolatey.
Melt the chocolate: Stir in the chopped chocolate. Keep whisking over low heat until all the chocolate melts and the hot chocolate turns glossy and thickened slightly.
Flavor it: Remove from heat. Stir in the vanilla and peppermint extracts.
Start with less peppermint, taste, and add a few more drops if you want a stronger mint finish.
Serve: Pour into mugs. Top with whipped cream, mini marshmallows, and a sprinkle of crushed candy canes if you like. Serve hot.