Preheat oven to 350 degrees F.
If using fresh chestnuts, cut a small cross in the rounded side of each chestnut, penetrating the thick shell. Place the chestnuts in a large pot and cover generously with water. Bring to a boil and simmer until they can be squeezed out of their shells easily, 15 to 25 minutes, depending upon the size and freshness of the nuts. Drain and peel, removing both the hard outer shell and the inner paper lining.
Place the chestnuts on a baking sheet and toast, about 10 minutes. Allow to cool. You should have about 3/4 cup of shelled chestnuts. If using frozen, canned, or jarred chestnuts, proceed from here.
Place the chestnuts in a food processor with 1 tablespoon of the sugar and process to a paste. You should have about 3/4 cup of paste.
In a bowl set over a saucepan of simmering water, combine the chocolate and the water. Cook until melted and smooth, stirring constantly. Beat in the egg yolks and whisk until thickened slightly, 2 to 3 minutes. Remove from the heat and fold in the chestnut paste and brandy. Cool completely.
In a medium bowl, whip the egg whites until foamy. While continuing to whip the whites, add the remaining 4 tablespoons of sugar little by little and continue to beat until stiff peaks form. Fold half of the whites into the chocolate mixture to lighten it, then gently fold in the remaining whites.
In a medium bowl, whip the heavy cream until it begins to thicken. Sprinkle in the confectioners' sugar and continue to beat until soft-stiff peaks form. Remove 1/2 cup of the whipped cream and set aside for garnishing the mousse. Gently fold the remaining whipped cream into the chocolate mixture.
To serve, spoon the mousse into decorative dessert bowls or coupes. Garnish with the sweetened whipped cream and chocolate shavings.