Chop the dark chocolate into small, even pieces to help it melt quickly and smoothly.
Pour the milk into a saucepan and set over medium-low heat, gently warming it until it begins to steam, about 3-4 minutes, with a soft bubbling sound.
Add the chopped chocolate to the warm milk and stir continuously with a silicone spatula. Watch as the chocolate melts into a smooth, glossy mixture, about 3-5 minutes.
Stir in the sugar or honey until fully dissolved, and continue stirring to keep the mixture velvety and evenly combined.
Remove the saucepan from heat and add a small splash (about 1/4 teaspoon) of peppermint extract. Taste and add more if desired for a brighter mint flavor.
Pour the hot chocolate into mugs, filling them just below the rim, and leave a little space for toppings.
Top each mug generously with whipped cream, creating a fluffy, creamy layer that contrasts beautifully with the smooth chocolate.
Garnish with crushed peppermint candies or a whole peppermint stick for a festive, crunchy finish.
Serve immediately while steaming hot, and enjoy the delightful combination of velvety chocolate and refreshing peppermint.