Pour the milk into a small saucepan and warm over low heat, watching for tiny bubbles forming at the edges—this takes about 5 minutes.
Add the cocoa powder and sugar to the warm milk, then whisk vigorously until the mixture is smooth, dark, and fragrant—about 1-2 minutes.
Stir in the peppermint extract, then continue whisking until evenly combined and the aroma of peppermint is prominent.
Increase the heat to medium-low and warm the mixture until steaming, about 2-3 minutes, but do not let it boil—keep an eye out for gentle steam and a rich aroma.
Pour the hot chocolate into mugs carefully, ensuring an even, frothy top with a glossy appearance.
Top each mug generously with whipped cream, then sprinkle crushed peppermint candies over the top for a festive finish.
Serve immediately while hot, enjoying the creamy texture, invigorating peppermint aroma, and the contrasting crunch of candies.