In a small saucepan, heat the milk and cocoa powder until steaming, then turn off the heat and stir in the chopped chocolate. Warm for 5 minutes, stirring frequently, until the chocolate melts. Stir in the vanilla extract.
In your favorite festive mug, combine the vodka, espresso, Kahlua, and simple syrup. Pour in the hot chocolate and whisk to combine.
Top your mug with whipped cream and chocolate curls, and lots of each!