Heat oven to 350°F. Coat a 9 x 2-inch round cake pan with nonstick vegetable-oil cooking spray. Line bottom of pan with waxed paper; coat paper with cooking spray; dust with flour; shake out excess flour.
In a large microwave-safe bowl, combine the chocolate and butter. Microwave on high (100%) power for 1 minute to melt; stir until smooth. Microwave for another 30 seconds, if necessary, to melt the mixture.
In a food processor, combine the drained chestnuts, heavy cream, brandy, egg yolks, 3/4 cup sugar, the flour and vanilla. Process until the mixture is smooth. Scrape into melted chocolate mixture; stir until blended and smooth.
In medium-size bowl, with an electric mixer on medium speed, beat together the egg whites and salt until foamy. Gradually beat in the remaining 1/4 cup sugar until stiff but moist peaks form. With a whisk, stir a quarter of the beaten egg whites into the chestnut mixture to lighten. Then fold in the remaining beaten egg whites until no white streaks remain. Spread the batter evenly into the prepared cake pan.
Bake the cake in the 350°F oven for 60 minutes or until a toothpick inserted in the center of the cake comes out clean. Transfer the cake pan to a wire rack. Carefully run a thin, long spatula around the inside edge of the pan to release the cake from the pan. Let the cake cool in the pan for 20 minutes.
Invert the cake pan onto the wire rack; carefully remove the pan. Peel off the waxed paper from the bottom of the cake. Let the cake cool completely on the wire rack.
In a medium-size bowl, stir together thawed whipped topping, cocoa powder and instant espresso powder until blended; reserve 2/3 cup of the topping for decoration.
Frost the sides and top of the cooled cake with the topping, spreading evenly and swirling. Spoon the reserved 2/3 cup of topping into a pastry bag fitted with a decorative tip. Decorate the cake as desired. Refrigerate the cake until serving time. Let the cake come to room temperature before serving.