Place the chestnuts and sugar in a saucepan and add enough milk to cover. Simmer until the chestnuts are quite tender and the milk fully absorbed by the chestnuts. Purée the chestnuts in a food processor or pass them through a food mill. Transfer to a large mixing bowl.

Add the cocoa, vanilla and rum to the chestnut purée, along with the sugar. Mix everything well with spatula until perfectly smooth and homogeneous, with a texture and feel akin to bread dough. (If you find it is a little too stiff, you can add a bit more rum to loosen it up.) Let this mixture rest in the fridge for at least an hour, as long as overnight.

Now it's time to make your 'mountain': Get a food mill or potato ricer, add the chestnut mixture (or as much as it will hold in one go) and turn or squeeze the handle until the mixture comes out the other end in strands. Let the stands fall onto a serving plate or, if making individual servings, the dessert bowl. As you proceed, the stands will pile up and form a mound resembling a mountain. Let that sit while you proceed to the next step. (You can also make the dish this far several hours ahead of time.)

Now make your 'snow' by whipping the cream with the confectioner's sugar until it forms soft peaks. Don't let the cream get too stiff.

Finally, finish off your dish by pouring the whipped cream over the chestnut and chocolate mountain. The cream should trickle down the sides, making an effect that looks like a snow-topped mountain. Serve immediately, or when you're ready to eat.