Pour the milk into a medium saucepan and warm over medium heat until it begins to steam, about 3-4 minutes. Keep an eye on it to avoid boiling.
Add the chopped dark chocolate to the warm milk, then whisk continuously until the chocolate melts completely and the mixture is smooth and glossy, about 2 minutes.
Stir in the sugar if using, and continue whisking until dissolved. Remove the saucepan from heat.
Gently stir in the peppermint extract, starting with a quarter teaspoon, then taste and add more if you desire a stronger mint flavor.
Crush the candy canes into small bits using your hands or a rolling pin, then stir most of them into the hot chocolate, allowing the peppermint flavor and texture to meld as they melt slightly.
Pour the hot chocolate into mugs, then top with whipped cream and sprinkle with additional crushed candy canes for a festive touch.
Serve immediately while warm, enjoying the smooth, velvety texture combined with the cool minty burst and festive garnishes.