Pour the milk into a small saucepan and set it over medium-low heat. Heat slowly, whisking occasionally. Do not let the milk come to a full boil.
After 4 to 6 minutes, or once steam begins rising from the surface, remove the pan from the heat.
Add the chopped chocolate to the hot milk, stirring continuously until completely melted, about 1 to 2 minutes.
Return the pan to medium heat and bring the hot chocolate to a gentle simmer for 2 to 4 minutes, whisking continuously until it reaches your desired thickness.
Turn off the heat, then stir in the brown sugar and vanilla extract. Mix until the sugar is fully dissolved.
Pour into a glass or cup and top with whipped cream and chocolate shavings. Enjoy!