Pour the milk into a medium saucepan and set over medium heat. Warm gently, watching for steam and small bubbles around the edges, which indicates it’s hot but not boiling.
Whisk in the cocoa powder and sugar, stirring constantly to dissolve and create a smooth, glossy mixture. Continue until the cocoa and sugar are fully incorporated and the mixture is steaming with no lumps.
Remove the saucepan from heat just as the mixture begins to bubble gently, ensuring it stays creamy and smooth.
Stir in the peppermint extract and a pinch of salt, giving the hot chocolate a bright, minty aroma and a balanced flavor. Mix well to evenly distribute the peppermint.
Pour the hot chocolate into mugs, filling them evenly. Observe the rich, velvety texture and deep chocolate color.
Optional: Top with whipped cream and crushed peppermint candies for added texture and festive presentation.