Pour the milk into a small saucepan and warm it over medium heat, watching it carefully until you see gentle steam and tiny bubbles around the edges, but no boil.
Add the chopped dark chocolate to the warm milk, then whisk continuously until the chocolate is fully melted and the mixture is smooth and glossy.
Gently tear the fresh peppermint leaves and add them to the hot chocolate, allowing them to steep for about 2 minutes to release their fragrant oils.
Remove the peppermint leaves from the hot chocolate, then stir in the sugar and vanilla extract if using, tasting and adjusting the sweetness as desired.
Pour the hot peppermint chocolate into mugs, watching the rich, velvety liquid swirl and steam rises as you fill each cup.
Optional: top with whipped cream and garnish with a sprig of fresh peppermint or a candy cane for a festive touch before serving.