MAKE THE CAKE (Makes one 9-by-13-inch rectangle)
1 Prepare the pan and preheat the oven: Heat the oven to 350F. Butter a 9-by-13-inch cake pan, dust the pan with flour, and tap out the excess. (Or line the pan with parchment paper and butter the paper.)
2 Combine the dry ingredients: In a bowl, sift together the flour, baking powder, baking soda, and salt.
3 Melt the chocolate and cocoa powder: Put the bittersweet or semisweet chocolate and the cocoa in a heatproof bowl and pour the boiling water over top. Let it sit for a minute, then stir. The chocolate might seem to seize at first, but you’re adding enough liquid that it should thin out as you stir.
4 Begin making the batter: In a stand mixer a beater attachment or with an electric mixer, beat the butter on medium speed for 2 minutes. Add the brown and granulated sugar and beat for 2 minutes more.
Add the eggs, one at a time, followed by the yolks and vanilla. Beat only until incorporated, scraping down the sides of the bowl often. The mixture may look curdled; that’s OK.
5 Finish mixing the batter: With the mixer on low speed, blend in the melted chocolate and cocoa powder. Blend in a third of the flour mixture, followed by half of the sour cream. Blend in another third of the flour, followed by the remaining sour cream. Finish with the last third of the flour, blending just until the dry flour is no longer visible. The batter will be fairly thick.
7 Transfer the batter into the pan: Scoop the batter to the prepared pan, spreading it evenly and smoothing the top. Tap it once hard on the counter to settle any air pockets.
8 Bake the cake: Bake the cake for 40 to 45 minutes, or until a skewer inserted into the center of the cake comes out clean clean and the top springs back when pressed with a fingertip.
Transfer the pan to a wire rack for the cake to cool. Allow the cake to cool completely before frosting. (At this point, the cake can also be covered and stored at room temperature for 24 hours before frosting and serving.)
MAKE THE FROSTING (Makes about 1 1/2 cups)
1 Melt the chocolate: Put the 8 ounces of semisweet chocolate in a heatproof bowl and set the bowl over a saucepan of hot but not boiling water. The bowl should not touch the water. Let the chocolate melt, stirring occasionally with a rubber spatula. Remove the bowl from the water and wipe the bottom dry. Cool, stirring often, for 5 minutes.
2 Add the sour cream gradually, stirring well with the rubber spatula.
3 Add the powdered sugar gradually, then the vanilla and salt. Stir until the mixture is smooth and thickens like frosting.
FINISH THE CAKE
1 Frost the cooled cake: When the cake is completely cool, use an offset icing spatula or the back of a spoon to spread the frosting on the cake, making a swirl or striped pattern, if you like.
To make “stripes” with frosting, as I have done, spread the frosting on the cake and smooth the top with an offset spatula. Wipe off the excess frosting on the spatula. Set the cake so one short side is closest to you. Holding the spatula at a 45-degree angle to the frosting, and working horizontally across the cake, start at the end farthest from you. Move the spatula from left to right across the cake. Immediately move the spatula down half an inch, and sweep it from right to left across the cake. Continue in this way – right to left, left to right, moving down half an inch – until the entire cake looks striped. The long sides will be scalloped with frosting.
To make frosting swirls, spread the frosting on the cake and use the back of a spoon to make slight indentations all over the cake, which will look swirled when you’re done.
2 Let the cake sit for 30 minutes before cutting into squares.