Pour the milk into a small saucepan and set over low to medium heat. Warm gently until you see steam rising, but avoid boiling.
Add the cocoa powder and sugar to the warm milk. Whisk vigorously to dissolve everything completely, creating a smooth, chocolatey mixture with a slightly frothy surface.
Stir in the peppermint extract, and continue to whisk for another minute to infuse the aroma into the hot chocolate. The mixture should be fragrant and uniform in color.
Remove the saucepan from heat once the mixture is hot and steaming, but not boiling. Carefully pour the hot chocolate into mugs.
If you like, top each mug with a dollop of whipped cream for extra richness, and garnish with a peppermint stick or crushed candies for a festive touch.
Serve immediately while warm, allowing the peppermint aroma to fill the room and the velvety texture to delight your senses.