Preheat oven to 350ºF/180ºC. Grease and flour a 8 x 8 baking dish; set aside.
In a mixing bowl, add all-purpose flour, cocoa powder, baking powder and salt; whisk until very well blended and set aside.
In another bowl, combine beaten eggs and melted butter, whisking constantly. To this, add brown sugar, vanilla sugar and vanilla extract; stir until combined.
Add egg mixture to dry ingredients and also some chopped walnuts if using; stir well to blend.
Transfer batter to the prepared baking dish and level it out using a spatula. Gently tap the dish on the counter to compact the batter and transfer to the preheated oven. Bake for 25 minutes or until a cake tester inserted in the center comes out clean with just a few crumbs.
Remove from the heat and allow to cool off.
Meanwhile, in a mixing bowl, combine sifted icing sugar, sifted cocoa powder, melted butter, corn syrup and pure vanilla extract.
While blending the ingredients, pour in some heavy cream a little at a time until desired consistency is reached.
When the cake is cool enough but still slightly warm which makes it easier to do, add the icing and spread it out evenly with a spatula.