To prepare chestnut purée, process chestnuts in a food processor until finely ground. Add 1 tablespoon butter; process until blended to measure 2 1/2 cups. (Add a few more chestnuts and puree, if necessary.) Cover and chill.
To prepare filling, combine 1/3 cup sugar, water, coffee granules, and 2 egg yolks in a food processor; process until smooth. Add chocolate; process until blended. Add 1/4 cup butter and 1/2 teaspoon vanilla; process until blended. Add 1/2 cup chestnut puree; process until smooth. Cover and chill 8 hours.
Preheat oven to 325°.
Beat 6 egg yolks in a large bowl at high speed of a mixer for 2 minutes. Gradually add 1 1/2 cups sugar, beating until thick and pale (about 5 minutes); beat in 1 1/2 teaspoons vanilla until blended. Gently fold in 2 cups chestnut purée. Beat egg whites at high speed of a mixer until stiff peaks form using clean, dry beaters (do not overbeat). Gently stir one-fourth of egg whites into chestnut mixture; gently fold in remaining egg whites. Pour batter into 2 (8-inch) round pans coated with cooking spray. Bake at 325° for 50 minutes. Cool completely on wire racks. Loosen cake layers by using a knife or narrow spatula. Place one cake layer on a plate; spread with filling, and top with other cake layer.
Spread whipped topping evenly over top and sides of torte. Chill 2 hours before serving.