Pour the milk into a medium saucepan and place over low to medium heat. Warm gently until it begins to steam, with tiny bubbles forming around the edges, and you can smell the warm milk aroma.
Add the sifted cocoa powder and sugar to the warm milk. Whisk continuously to dissolve the powders completely, creating a smooth, chocolatey mixture. You should hear a gentle bubbling as the mixture heats.
Stir in the finely chopped dark chocolate, whisking until it melts completely into the hot cocoa mixture, resulting in a glossy, velvety surface with a rich aroma filling the air.
Remove the saucepan from heat. Add the drops of peppermint extract, starting with a few and tasting to reach your desired level of minty freshness. Stir well to incorporate.
Pour the hot chocolate through a fine-mesh sieve into mugs to strain out any lumps or remaining peppermint bits, ensuring a silky-smooth texture.
Top each mug generously with homemade whipped cream, creating a fluffy, inviting cloud on top of the rich chocolate.
Sprinkle crushed peppermint candies over the whipped cream for a festive crunch and burst of peppermint flavor.
Garnish with a small sprig of fresh mint or a dusting of cocoa powder if desired. Serve immediately with a cozy spoon or straw, and enjoy the warm, minty aroma with each sip.