Heat 1 cup of the milk in a medium saucepan over low heat. Add chocolate, stirring constantly, until melted. When the chocolate has melted, increase heat to medium and add the remainder of the milk while whisking rapidly. Do not allow to boil. Add 1 ounce peppermint schnapps to each (of 6) mugs. Pour the hot chocolate mixture into the mugs and top with whipped cream (if desired). Add mini candy canes (or peppermint sticks) to each mug before serving, if desired.
This is an excellent hot cocoa recipe on its own, without alcohol. But the peppermint schnapps adds an extra kick. Hang mini candy canes on the rim of the cup for a holiday garnish that doubles as a swizzle stick. Serve topped with whipped cream.