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.
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).
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.)
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.
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.
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