Preheat oven to 350° F. and grease a 9-inch round cake pan. Line the bottom of the pan with a round of parchment paper. Set aside.
In a medium bowl, whisk together the flour, cocoa, baking powder, baking soda and salt. Set aside.
In a large bowl with an electric beater or in the bowl of a stand mixer fitted with a paddle attachment, beat the butter with the brown sugar on medium high for about 3 minutes. Add the eggs one at a time, beating well after each addition and scraping down the sides of the bowl as needed. Add the vanilla and beat in. Remove the bowl from the stand mixer (if using) and using a wooden spoon, stir in 1/3 of the flour mixture, then add 1/2 of the stout. Add another 1/3 of the flour, followed by the rest of the stout and finally the last 1/3 of the flour mixture. Stir until batter is evenly mixed. Pour batter into prepared pan and level top.
Bake in pre-heated oven for about 40 minutes or until a tester inserted in the middle comes out clean. Allow to cool in pan for 10 minutes, then run a knife around the outside and remove from pan. Peel parchment from the bottom of the cake and then place cake on cooling rack to cool completely. Can be made ahead. Once cooled, wrap in plastic wrap and refrigerate or freeze.
For the frosting: In a small saucepan, combine the sugar and water. Bring to a boil over medium-high heat and once boiling, allow to boil, stirring occasionally, until sugar is dissolved and mixture is clear. Remove from heat and stir in vanilla. (I like to pour the mixture into a glass measuring cup at this point, to make it easier to add to the egg whites.)
In a large bowl or the bowl of a stand mixer fitted with a whisk attachment, beat the egg whites and cream of tartar on high until soft peaks form when beater is lifted from the mixture. With mixer running on medium, slowly start adding the sugar mixture in a slow stream down the side of the bowl. Increase beater to high and beat until stiff peaks form, about 2-3 minutes.
To assemble: Place cooled cake on cake plate or stand. Pile frosting on top and using the back of a kitchen teaspoon, move towards the edges and create swirls. If desired, you can lightly toast the frosting with a kitchen torch.
Be sure to read the "Cook's Notes" in the original post, for more tips, options, substitutions and variations for this recipe!