For the chocolate stout cake: Preheat the oven to 375 degrees F. Grease a 9-inch round cake pan with the room temperature butter, then line it with parchment paper.
Whisk together the flour, granulated sugar, cocoa powder, baking soda and salt in a large bowl. Combine the stout, melted butter, vanilla extract and vinegar in a separate large bowl and mix well. Add the wet ingredients to the dry ingredients and mix until just combined. Pour the batter into the prepared pan and bake until a toothpick inserted in the center comes out with a few crumbs, 30 to 40 minutes. Transfer to a wire rack and let cool completely.
For the Irish cream frosting: Beat the cream cheese in a large bowl with an electric mixer until smooth, about 1 minute. Add the confectioners' sugar and mix on low speed until smooth and combined. Add 1 tablespoon of the Irish cream liqueur and mix until smooth and spreadable, 2 to 3 minutes. You can add another tablespoon of liqueur if needed to get the right consistency--it should be smooth but still thick enough to spread.
Remove the cake from the pan and spread the frosting on top. Use a vegetable peeler to shave the chocolate over the top. Serve.