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.
Serve in small cups fortified with a jigger (1 1/2 ounces) of peppermint schnapps and a dollop of whipped cream. Add a peppermint stick to stir. If transporting in a thermos, heat thermos up with hot water to get it warm for 1 minute. Pour water out and fill with hot chocolate.