Pour the milk into a saucepan and set over medium heat. Warm until it begins to steam, with small bubbles forming around the edges, and the aroma of warm milk fills the air.
Add the cocoa powder and brown sugar to the warm milk. Whisk vigorously to dissolve the cocoa and sugar completely, creating a smooth, glossy mixture with no lumps, and a rich chocolate scent filling the kitchen.
Stir in the chopped dark chocolate and continue whisking until the chocolate melts and the mixture thickens slightly, becoming velvety and uniform in texture. You should hear a gentle simmer and see the chocolate fully incorporated.
Remove the saucepan from heat. Add the peppermint extract, starting with half a teaspoon, and whisk well. Taste the hot chocolate and add more peppermint if you want a stronger mint flavor.
Pour the hot peppermint chocolate into mugs, filling them generously. The mixture should be smooth, steaming, and inviting, with a deep chocolate hue and a faint peppermint aroma.
Top each mug with a generous swirl of whipped cream. Sprinkle crushed peppermint candies over the whipped topping for a festive, crunchy finish.
Serve immediately while steaming hot, and enjoy the velvety, minty-sweet flavor with every sip.