Combine the chestnuts with the milk, sugar, and rum in a saucepan and cook over low heat until the chestnuts are soft, 20 to 30 minutes. Cool, then purée in a food processor until smooth. Keep the chestnut purée at room temperature until ready to use.
Preheat the oven to 212 degrees F.
Whip the egg whites slowly, and carefully add the granulated sugar. When the whites reach a hard peak, fold in the sifted confectioners sugar. Line a large baking sheet with parchment paper and fit a pastry bag with a round ½-inch tip. Mark ten 2 ½-inch circles and pipe with the meringue. Bake for 1 hour. The meringue disks can be made up to a week in advance and stored in a container in a cool dry place.
Whip the cream until almost stiff. Add the sugar and vanilla, then beat until cream holds its peaks.
Place a meringue disk on a serving plate. Fill a pastry bag with chestnut purée. Using a circular motion, build layers of purée around the circumference of the meringue disk from edge to center, developing it into a mound or mountain-like shape. Cover the mound by piping lines of whipped cream from the bottom outside edge to the top center. Use a slight curve in the lines to create the look of a snowcapped mountain peak, such as Mont Blanc, for which the dessert is named.