Pour the milk into a saucepan and set over medium-low heat. Warm gently, watching closely until it begins to steam and tiny bubbles form around the edges, about 3-5 minutes.
Add the chopped chocolate or chocolate chips to the warm milk. Stir continuously with a wooden spoon or spatula until the chocolate is fully melted and the mixture is smooth and glossy, about 2-3 minutes.
Stir in the peppermint extract or crushed peppermint leaves. Let the mixture simmer gently for another 2 minutes to infuse the mint flavor, filling the kitchen with a fragrant aroma.
Taste the hot chocolate and add sugar if needed to reach your preferred sweetness. Stir well to combine.
Remove the saucepan from heat. Using a fine mesh sieve, strain the hot chocolate into mugs to remove any peppermint leaves or chocolate bits, ensuring a silky smooth texture.
Top each mug with a generous dollop of whipped cream. Garnish with a sprig of fresh mint or a peppermint candy for a festive touch.
Serve immediately while steaming hot. Enjoy the cozy aroma and smooth, minty flavor with every sip.