Preheat the oven to 350°F and thoroughly grease an 18x13-inch jelly roll pan. In a medium saucepan set over medium heat, combine the butter, water, and cocoa powder. Bring the mixture to a rolling boil. In the meantime, whisk together the flour, sugar, baking soda, and salt in a separate large mixing bowl. In a smaller bowl, blend the sour cream, eggs, and vanilla extract until smooth. Add the sour cream mixture to the dry ingredients and stir until incorporated. Carefully pour the hot chocolate mixture into the batter and stir until the consistency is uniform and smooth. Transfer the batter into the prepared pan, spreading it evenly with a spatula. Bake for 15–20 minutes, checking for doneness around the 15-minute mark.

While the cake bakes, prepare the frosting. In a saucepan, heat the butter, milk, and cocoa powder over medium heat until the mixture boils. Remove from heat and immediately whisk in the powdered sugar, using a hand-held mixer if necessary to ensure a smooth texture. Pour the warm frosting over the freshly baked, still-hot cake. Use a spatula to evenly spread the icing across the surface. Let the frosting set for approximately 10 minutes before serving.