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For the cake: Lightly grease a 9-by-13-inch cake pan with butter. Combine the flour, sugar and baking soda in the bowl of a stand mixer or mixing bowl. Combine the butter, cocoa and 1 cup water in a medium saucepan and bring to a boil while mixing. Add the cocoa mixture to the flour mixture, along with the buttermilk eggs and vanilla, and beat for 3 minutes. Pour into the prepared cake pan and bake until firm to the touch and a toothpick inserted in the center comes out clean, 30 to 40 minutes. Let cool 10 minutes in the pan before turning out on a rack to cool completely.
For the frosting: Make the frosting as soon as the cake is out of the oven. Combine the butter, buttermilk and cocoa in a small saucepan and bring to a boil while mixing. Combine with the confectioners' sugar in the bowl of a stand mixer and beat until smooth. Spread the frosting evenly on the cake.