Pour the heavy cream and milk into a saucepan and warm over medium heat until just steaming, with small bubbles forming around the edges. You should be able to hear a gentle simmer.
Add the chopped dark chocolate or chocolate chips to the warm milk mixture. Stir continuously with a whisk, creating a gentle sizzle, until the chocolate is fully melted and the mixture is smooth and glossy.
Stir in the sugar and peppermint extract, tasting as you go. Adjust the peppermint to your preference—start with a small splash and add more if you want a stronger minty flavor. Continue whisking until everything is well combined and heated through.
Remove the saucepan from heat once the mixture is steaming and smooth. Check the temperature—it's ready when it's hot but not boiling. Pour the hot chocolate into mugs, filling them generously.
Top each mug with a dollop of whipped cream and garnish with a peppermint candy or sprig of mint for a festive touch. Serve immediately while warm and fragrant.