Pour the milk into a 2-quart, heavy-based saucepan and set over medium heat, stirring occasionally. Heat the milk until it is gently steaming, but not simmering.
Meanwhile, finely chop the bar with a sharp knife.
Once the milk is gently steaming, remove the pan from the heat and dump in the chopped chocolate. Whisk continuously until it is melted and the mixture is smooth. Be patient; this will take a couple minutes.
Place the saucepan over medium heat and bring to a gentle boil and cook until thickened, about 5 minutes. Taste and whisk in the brown sugar if desired.
Serve hot in small cups, garnished with a sprinkle of coarse sea salt and a dollop of Chantilly cream. Alternatively, let it cool completely to room temperature and refrigerate for up to 3 days. Reheat in the microwave or on the stovetop over gentle heat.
Pour the chilled whipping cream into a bowl and add the sugar and vanilla extract. Whip with a handheld electric mixer on medium speed until the cream is stiff. It shouldn't fall out of the bowl if it is turned upside down.
Place a dollop of the Chantilly cream in each cup. Chantilly cream is best when enjoyed right away, but you can store leftovers covered tightly in the fridge for a day.