Preheat oven to 350 degrees F (175 degrees C). Line a 9-inch cake pan with parchment paper; grease parchment paper.

Combine stout beer and butter in a large saucepan over medium-low heat; heat until butter melts and beer looks glossy, 3 to 5 minutes.

Remove beer mixture from heat; whisk in sugar and 3/4 cup plus 2 tablespoons cocoa powder until dissolved and smooth. Whisk in 1/2 cup plus 2 tablespoons sour cream. Mix in eggs and vanilla extract until well blended. Add baking soda; mix in flour gradually until batter thickens and no lumps remain, about 5 minutes.

Pour batter into the prepared pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 40 minutes. Cool cake before serving, at least 4 hours.