Preheat your oven to 350° and line an 9x9 pan with parchment paper leaving 2-inches of over hang on all sides.
In a small sauce pan, simmer the beer until it's reduced to 1/3 of a cup. Set aside and let cool.
Bring a pot (filled 1/4 of the way) with water up to a boil. Reduce to a simmer and place a bowl over top, the water should not touch the bottom of the bowl. Add the chocolate and the butter. Stir until both are melted and combined. Remove and let cool slightly.
In a large bowl whisk together the eggs, vanilla, sugars and espresso powder. Gradually add in the slightly cooled chocolate/butter mixture and the reduced Guinness, whisking to combine. The batter should thicken.
Lastly, add in the flour and salt. Gradually stir to combine until no traces of flour remain, do not over mix.
Pour the brownie batter into your prepared pan. Slide the pan onto the middle rack of your preheated oven and bake for 30 to 40 minutes or until desired doneness. Let cool completely before frosting.
FOR THE BROWN BUTTER STOUT FROSTING:
In a 10-inch skillet melt the butter on medium-low heat, stirring and scraping the sides until it browns and turns a deep golden color. Remove and let cool slightly.
Meanwhile, in a large mixing bowl, sift the powdered sugar. Pour in the brown butter and combine using an electric mixer.
Add in the stout and mix on low to combine. Add enough milk until you reach a desired consistency, about 2 to 3 tablespoons.
Spread frosting on cooled brownies, cut and serve.