Step 1

Preheat oven to 350 degrees. Butter a 9-by-13-inch baking pan. Line bottom with parchment; butter parchment. Stir together boiling water, cocoa, and chocolate; let cool 10 minutes. Meanwhile, whisk together flour, baking powder, baking soda, and salt in another bowl.

Step 2

Beat butter with sugar on medium-high speed until fluffy, 2 to 3 minutes. Beat in eggs, one at a time, beating to combine after each addition and scraping down sides of bowl as needed. Beat in vanilla, then chocolate mixture. Reduce speed to low; beat in flour mixture in two batches, alternating with sour cream, until just combined. Scrape batter into prepared pan, smoothing top with an offset spatula.

Step 3

Bake until a tester inserted in center comes out with a few moist crumbs, 35 to 40 minutes (top will not spring back). Let cool in pan 10 minutes. Invert onto a wire rack and remove parchment; let cool completely. Cake can be made up to 1 day ahead and stored at room temperature, wrapped in plastic.

Step 4

Spread buttercream evenly over top and sides of cake. Sprinkle with chocolate curls. (Leftovers can be refrigerated, wrapped in plastic, up to 3 days; return to room temperature before serving.)