Pour the milk into a saucepan and set over medium-low heat. Warm gently, watching for tiny bubbles to form at the edges, and listen for a slight simmer to develop.
Add the chopped dark chocolate or chips to the warm milk. Stir continuously with a whisk until the chocolate melts completely and the mixture becomes smooth and glossy, about 2-3 minutes.
Stir in honey or maple syrup along with vanilla extract. Continue whisking until the sweetener blends in and the mixture tastes balanced and fragrant.
Add a tiny drop of peppermint oil, then stir vigorously. Smell the mixture to check for a fresh mint aroma, and taste to ensure the flavor is to your liking. Adjust with more peppermint if desired.
Allow the hot chocolate to simmer gently for another minute, stirring constantly. If it begins to crackle or bubble too vigorously, lower the heat to prevent burning.
Remove the saucepan from heat and pour the hot chocolate into mugs. For extra smoothness, you can strain the mixture through a fine sieve to catch any lumps or foam.
Top with whipped cream, a peppermint candy, or a sprinkle of cocoa powder for a festive finish. Serve immediately while warm and enjoy the bright, velvety sip.