This dessert is mounded into the shape of a mountain and covered with whipped cream. Twenty ounces of frozen chestnuts can be substituted for two pounds of fresh.
Using a small paring knife, cut an X into one side of each chestnut. Fill a saucepan with water; add chestnuts and salt; bring to a boil. Reduce heat to medium; cook until tender, about 20 minutes. Remove 2 or 3 at a time from water; remove the outer shells. Peel and scrape off inner skins.
Combine nuts, milk, 3/4 cup sugar, butter, and vanilla bean and scrapings in a saucepan; bring to a simmer over medium heat. Cook until tender and falling apart, 30 to 35 minutes. Drain, reserving milk. Remove vanilla bean; set nuts aside to cool.
Pass chestnuts through a food mill into a bowl. Stir in rum and 5 to 6 tablespoons of the reserved milk until well combined. Place 1/3 cup of chestnut mixture in a potato ricer fitted with large hole blade. Press and squeeze strands of the mixture into bowls. It should look like a small mound of spaghetti. Repeat with remaining mixture.
Whip cream with remaining 1/4 cup sugar. Fill a pastry bag fitted with a large star tip. Pipe out cream covering the exposed surface. Dust with cocoa, and serve.