Break the chocolate into pieces and place in a sauce pot. Set the pot over medium-low heat and add the milk, cream, brown sugar, and vanilla extract.
Stir until the chocolate comes to a low simmer and the drinking chocolate is completely smooth and slightly thickened.
The drinking chocolate should be substantially thicker than classic hot chocolate, but not as thick as chocolate fondue. If it is too thin, simmer a little longer. If it is too thick, add another 1/4 cup milk. Ladle into mugs and enjoy immediately.