In a medium saucepan, bring 1 ½ cups of milk to a gentle simmer over medium heat, then add the roasted, peeled chestnuts. Stir occasionally and let it simmer for about 5 minutes. Transfer mixture to a high-speed blender and blend until smooth.

Return the saucepan to the stove, pour in the remaining 1 ½ cups of milk and ½ cup of heavy cream. Heat over medium-low until it simmers. Stir in the chopped dark chocolate until melted, then add cocoa powder, sugar, and vanilla extract; whisk until velvety.

Pour the blended chestnut milk into the chocolate mixture in the saucepan. Whisk together until fully combined, and heat gently for an additional 2–3 minutes.

In a separate mixing bowl, beat 1 cup of heavy cream on medium speed until soft peaks form. Gradually add icing sugar, vanilla extract, and dark rum; continue beating until stiff peaks form.

Pour the hot chocolate into warm mugs, top generously with the boozy whipped cream, and grate some dark chocolate on top for garnish. Serve immediately.