Pour the milk into a saucepan and set over medium-low heat, warming gently until it begins to steam, about 3-4 minutes, with a soft simmer but no boiling sounds.
While the milk warms, sift in the cocoa powder into the pan to prevent lumps, then whisk vigorously until the mixture is smooth and heated through, about 1-2 minutes.
Add the sugar gradually, tasting and adjusting as needed, whisking until dissolved and the mixture is uniformly velvety.
Crush the fresh peppermint leaves slightly with your fingers or a muddler, then add them to the warm milk.
Simmer the milk with the peppermint leaves for about 3-5 minutes, allowing the flavor to infuse and the aroma to fill the kitchen.
Remove the saucepan from heat, then strain out the peppermint leaves into a bowl or directly into mugs, pressing gently to extract all the flavor.
Stir in the vanilla extract, and if desired, add a few drops of peppermint extract for a more intense minty flavor.
Pour the hot chocolate into mugs, then top generously with whipped cream and garnish with a fresh peppermint sprig for a festive touch.
Serve immediately while warm, enjoying the fragrant aroma and the contrasting cool peppermint burst with each sip.