Pour the milk into a small saucepan and warm it over low heat, watching for tiny bubbles to form around the edges, which takes about 5 minutes.
Add the chopped chocolate or chocolate chips gradually into the warm milk, stirring constantly with a whisk or spoon until completely melted and the mixture is smooth and glossy, about 3-4 minutes.
Stir in a few drops of peppermint extract, tasting and adjusting the amount until the refreshing mint flavor is balanced with the rich chocolate.
Pour the hot chocolate into mugs carefully, then top each with a generous dollop of whipped cream.
Sprinkle crushed peppermint candies over the whipped cream for a festive crunch and vibrant color.
Serve immediately while warm, enjoying the fragrant aroma and smooth, glossy texture of your peppermint hot chocolate.