In a small saucepan combine 1 ½ cups of the milk with the chestnuts. Bring to a simmer over medium heat. Turn off heat and carefully transfer to a blender. Set aside while you prepare the chocolate mixture.
Add the remaining 1 ½ cups milk along with the heavy cream to a saucepan and bring to a simmer over medium heat. Finely chop the dark chocolate and add to the milk mixture, stirring until the chocolate melts. Add the cocoa powder, sugar, and vanilla extract and whisk together.
When the chocolate has melted completely and the mixture is smooth, carefully pour into the blend along with the chestnut milk. Cover and blend the hot chocolate until completely smooth and frothy. Pour into a clean saucepan and reheat while preparing the whipped cream.
Boozy Whipped Cream
In a bowl beat the heavy cream with a hand mixer or in a stand mixer until soft peaks. Add the icing sugar, vanilla extract, and dark rum to the whipped cream and beat until fully combined.
Ladle the chestnut hot chocolate into mugs and top with the boozy whipped cream. Garnish with grated dark chocolate and enjoy!