Preheat the oven to 350°F (175°C) and grease a 9x13-inch pan. Sift and whisk together the flour, sugar, cocoa powder, baking soda, baking powder, and salt.

Add the eggs, buttermilk, vegetable oil, and vanilla to the dry ingredients and whisk until smooth, then stir in the hot water until a thin batter forms.

Pour into the prepared pan and bake for 30 to 35 minutes, until a toothpick comes out with a few moist crumbs. Cool completely on a wire rack.

Melt the butter in a saucepan over medium heat, whisk in the cocoa powder for 1 minute, add the milk and stir for 2 to 3 minutes until hot, then remove from heat and whisk in the sifted powdered sugar and vanilla until smooth and glossy.

Pour the warm icing over the cooled cake, spread to the edges, and let it set for 20 minutes. Slice into 12 squares and serve.