Pour the milk into a saucepan and warm over medium-low heat, listening for a gentle simmer and watching for steam rising, without letting it boil.
Add the cocoa powder to the warm milk and whisk continuously, breaking up any lumps until the mixture is smooth, fragrant, and slightly thickened.
Stir in the sugar and a pinch of salt, whisking until fully dissolved and the drink is glossy and uniform.
Add the peppermint extract and stir well, then let it simmer gently for about 2 minutes to infuse the peppermint aroma.
Remove the saucepan from the heat and strain the hot chocolate through a strainer into a pitcher or directly into mugs, removing any peppermint leaves or solids.
Pour the hot chocolate into warmed mugs, then top with whipped cream or marshmallows for extra richness and visual appeal.
Finish by sprinkling crushed candy cane or adding a small peppermint stick for a festive touch, then serve immediately while steaming hot.