Pour the milk into a saucepan and set over medium-low heat. Warm gently until tiny bubbles appear around the edges, and the milk is steaming but not boiling.
While the milk warms, whisk together cocoa powder and sugar in a small bowl to break up any lumps.
Add the cocoa mixture to the warm milk, whisking constantly until the cocoa and sugar are fully dissolved and the mixture is smooth and glossy.
Stir in the peppermint extract, starting with a small splash and tasting as you go to achieve your preferred level of minty brightness.
Continue whisking for another minute until the aroma of chocolate and peppermint is inviting and the mixture is heated through.
Pour the hot chocolate into mugs, then top generously with whipped cream. Finish by sprinkling crushed candy canes over the whipped cream for a festive touch.