Heat the milk and cream in a saucepan over medium heat, just until the edges are bubbling. Stir in the chopped chocolate and turn off the heat. Stir until the chocolate melts. Remove from the heat and let cool for 10 to 15 minutes.
Stir in the vanilla extract, then stick the pot in the fridge to cool completely, about 45 minutes.
Remove the pot and add the chocolate mixture to a blender. Add the schnapps and 2 cups of ice and blend until smooth and creamy. Pour evenly between 2 glasses and top with whipped cream and crushed peppermints.