Heat the oven to 350°F (175°C), with a rack in the middle. Lightly grease a 9-by-13-inch metal baking pan. Measure the ingredients before mixing; keep raw flour and eggs away from the frosting ingredients.

In a large bowl, whisk the flour, granulated sugar, cake cocoa, baking powder, baking soda and salt until evenly combined and free of cocoa lumps.

In a second bowl, whisk the eggs, cake milk, oil and cake vanilla. Add to the dry ingredients and whisk just until combined. Carefully pour in the hot water in two additions, stirring gently until smooth. The batter will be thin; do not add extra flour. Do not taste raw batter.

Pour into the prepared pan and bake for 30–40 minutes, starting to check at 30 minutes. The center should spring back lightly and a toothpick inserted in the middle should come out with a few moist crumbs, not wet batter. If the center is still wet, continue in 3-minute intervals.

Set the pan on a cooling rack and cool completely, about 90 minutes or longer if needed. Do not frost a warm cake.

For the frosting, beat the softened butter with a hand mixer for about 1 minute. Add powdered sugar, frosting cocoa, 2 tablespoons of the frosting milk and frosting vanilla. Beat on low until combined, then medium for 1–2 minutes. Add the remaining tablespoon milk a teaspoon at a time only as needed for a soft spreadable consistency.

Spread the frosting thinly over the cooled cake. Cut into 16 squares. Cover and refrigerate, then let individual slices stand briefly before serving. For best quality, use within 3–4 days or freeze wrapped portions for up to 2 months.