Set the oven to 350°F and grease a 9" x 13" baking pan. Coat the corners as well as the bottom to help the finished slices release.

Stir the hot coffee or water into the cocoa in a large bowl, then incorporate the oil and buttermilk. Work out any dry cocoa pockets so the mixture looks evenly dark and smooth.

Beat in the baking soda and salt for 30 seconds, blend in the vanilla and eggs, then add the sugar and flour and mix at medium speed for 2 minutes before scraping the bowl, beating another 30 seconds, and transferring the batter to the pan. The finished batter should look uniform, with no flour clinging to the bowl.

Bake for 33 to 35 minutes, rest the pan on a rack for 30 minutes, then freeze the cake for at least 1 hour. Before removing it from the oven, check that a gentle press rebounds and a slight gap has begun to form along the pan edges.

Whip the butter and salt until light, beat in the confectioners’ sugar and vanilla until fluffy, then add milk or half-and-half as needed, starting with 1 tablespoon, and fill a piping bag fitted with a 1/2-inch round tip. The icing should hold its shape while remaining smooth enough to pipe; refrigerate it for 5 to 10 minutes if it feels overly soft.

Pipe seven edge-to-edge buttercream strips across the cake’s shorter, 9-inch dimension, making each strip 1/2 inch in diameter with 1 inch between lines, then freeze for at least 30 minutes. Keep steady pressure on the bag for rounded ridges that become firm before coating.

Heat the buttermilk, corn syrup, butter, cocoa, and salt in a medium heavy-bottomed saucepan over medium heat, boil for 2 to 3 minutes, reduce to low, whisk in the confectioners’ sugar until dissolved, then remove from the heat, stir in vanilla, and transfer to a large glass measuring cup to cool. Listen for sharp snaps from the bursting bubbles during the boil, and wait until the finished glossy glaze feels only slightly warm before pouring.

Pour half the fudge over the frozen cake in steady passes, gently tipping the pan wherever the coating needs help reaching. Let the flowing glaze surround the buttercream ridges rather than dragging a knife across them.

Freeze the first coating for 5 to 10 minutes, pour on the remaining fudge, and refrigerate the cake for 20 minutes before cutting. If the reserved glaze stops flowing smoothly, microwave it in 5-second bursts with stirring between each burst.

Keep the finished cake covered in the refrigerator until serving. A chilled coating and firm buttercream centers make the slices easier to handle.