This is a seriously compelling piece of cake bakery: definitely an after-dinner dessert— and an elegant one at that—rather than afternoon tea, though I think I might be able to force a slice down with a cup of coffee at unscheduled moments in the day. Really, you need serve nothing with it: apart from anything else, it's not so sweet as to need the masking properties of cream.
Preheat the oven to 350°F.
Beat the chestnut purée with the butter, then add the vanilla, rum, egg yolks, and melted chocolate, blending well. I use my KitchenAid here, but an ordinary electric hand-held mixer would be fine-or even a bowl and wooden spoon. In another large bowl, whip the egg whites with the salt until they are foamy. Add the sugar gradually to form stiffer, glossy peaks, and then sprinkle the brown sugar over and either fold in or whisk in slowly. Fold the whites, gently but confidently, into the chestnut mixture, a third at a time.
Pour into the pan and cook for 45 minutes, until the cake has risen and is firm on top; it will look dry and cracked, but don't panic: it won't taste dry, and the cracks don't matter a damn.
Cool in the pan for 20 minutes, and then turn out on a rack.
When you want to eat it, dust with confectioners' sugar and serve with modest pride.