Preheat the oven to 350 degrees F (175 degrees C). Grease and flour a 10×15-inch cake pan.

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Combine butter, water, and cocoa powder for cake in a saucepan over medium heat. Cook until butter has melted and mixture comes to a boil. Remove from the heat and cool slightly.

Meanwhile, mix flour, sugar, and salt together in a large bowl. Beat buttermilk, eggs, baking soda, and vanilla extract together in a separate bowl.

Add melted butter mixture to the flour mixture, then add buttermilk-egg mixture and beat until thoroughly combined. Pour into the prepared cake pan.

Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 20 minutes. Remove cake from the oven and cool on a wire rack for 5 minutes.

Meanwhile, combine butter, milk, cocoa powder, and vanilla extract for the frosting in a saucepan over medium heat. Stir until butter melts, then stir in powdered sugar. Spread frosting over the warm cake.