Preheat your oven to 350°F (175°C). Grease an 18x13-inch baking pan thoroughly.

In a medium-sized saucepan over medium heat, combine the butter, water, and unsweetened cocoa powder. Stir until the mixture reaches a boil, then remove from heat.

In a separate large mixing bowl, combine the granulated sugar, all-purpose flour, baking soda, and salt. Mix well to ensure the dry ingredients are evenly distributed.

In another small bowl, whisk together the eggs, sour cream, and vanilla extract until smooth. Add this mixture to the dry ingredients, stirring until just combined.

Gradually pour the hot chocolate mixture from the saucepan into the batter, stirring continuously to create a smooth and uniform consistency. Make sure no lumps remain in the batter.

Pour the prepared batter into the greased pan, using a spatula to spread it evenly. Bake for 15 to 20 minutes, or until the cake is set. To check doneness, insert a toothpick into the center; it should come out clean or with a few moist crumbs.

While the cake bakes, prepare the frosting. In a saucepan, combine the butter, milk, and cocoa powder over medium heat. Bring this mixture to a boil, then remove from heat.

Gradually whisk in the powdered sugar until the frosting is smooth and free of lumps. For a smoother texture, use a hand-held mixer if necessary.

As soon as the cake is done, pour the hot frosting over the warm cake. Spread it evenly using a spatula, ensuring the icing covers the entire surface. Allow the frosting to set for about 10 minutes before slicing and serving.