Step 1

For the cake: Preheat the oven to 350 degrees F. Line a 9-by-2-inch round cake pan with parchment paper, and spray with cooking spray.

Step 2

Whisk together the flour, sugar, cocoa powder, baking powder, baking soda and salt in a medium bowl. Beat together the buttermilk, egg and vanilla in a measuring cup. Pour the wet ingredients into the dry, and beat with a wooden spoon until well combined, scraping the side and bottom of the bowl once or twice with a rubber spatula. Stir in the hot water until it is all absorbed and the batter is smooth (it will be thin).

Step 3

Pour the batter into the prepared pan, and bake until the top is shiny, the cake is just beginning to pull away from the side of the pan and a toothpick inserted in the center comes out with a crumb or 2 attached, about 20 minutes. Let the cake cool in the pan on a cooling rack for 20 minutes. Turn the cake out of the pan, remove the parchment paper and let the cake completely cool on the rack, right-side up, about 1 hour. (The cake will be thin, but don't worry-it will magically be OK once it's stacked with frosting.)

Step 4

For the frosting: Pulse the butter, cocoa powder, confectioners' sugar and salt in a food processor 10 times, stopping halfway through to scrape the bottom and side of the bowl with a rubber spatula. Add the corn syrup and vanilla, and pulse 5 times. Scrape the bottom and side of the bowl, drizzle in the chocolate and pulse 5 more times to combine. Transfer the frosting to a small bowl.

Step 5

To assemble: Cut the cooled cake into 4 even quarters. Frost the top of each quarter with about 2 tablespoons of the frosting (an offset spatula helps spread the frosting evenly). Stack the frosted quarters directly on top of each other on a serving plate, resulting in a 4-layer wedge of cake with frosting between layers and on top. Frost all 3 sides of the cake with the remaining frosting until completely covered. Let sit for 30 minutes. Halve and serve.