Preheat oven to 350 degrees F (175 degrees C). Grease and flour two 9 inch round layer cake pans.
Beat eggs and white sugar together with an electric mixer. Add oil, wine, flour, salt, baking powder, and 1 teaspoon vanilla. Beat for about 1 to 1 1/2 minutes.
Pour batter into pans and bake for about 30 to 35 minutes or until cake tests done.
Cool cakes in pans for 10 minutes. Turn them out on wire rack and let cool for at least 2 to 3 hours before frosting.
Using a cake slicer or a serrated knife cut each layer in half lengthwise. Arrange one layer on serving dish and spread with one third of the chestnut puree, top with second layer spread 1/3 of chestnut puree on top, top with third layer and spread remaining chestnut puree. Arrange last layer on top and set cake aside.
To Make Icing: Melt butter and chocolate together in the top part of a double boiler. Once melted remove from heat and whisk in cream, confectioners' sugar, and 1 teaspoon vanilla. Pour warm chocolate icing over top of cake and garnish with whole candied chestnuts.