Sour Cream Chocolate Cake with Coconut Frosting

TO MAKE THE CAKE:
Heat the oven to 300 degrees F. Line the bottoms of two 9-inch cake pans with kitchen parchment and set them aside (there's no need to grease the pans). Put the cocoa powder in the bowl of an electric mixer fitted with the whisk attachment. Pour the boiling water over the cocoa and whisk until smooth. Add the butter and sour cream and blend on low speed until the butter melts. Allow the mixture to cool for a minute if still very hot, and then add the eggs and vanilla and whisk until smooth. Cool for 10 minutes.

Meanwhile, sift together the cake flour, sugar, baking soda, and salt. With the mixer on low speed, add the dry ingredients a little at a time to the butter mixture, scraping down the sides once or twice. Increase the speed to medium and blend for another 3 minutes. Pour the batter into the prepared pans and bake until the center of the cake feels firm and the cake just barely begins to pull away from the sides of the pan, 50 minutes to 1 hour (begin checking after 45 minutes). Remove the cakes from the oven and let cool completely before frosting.

TO MAKE THE FROSTING:
Put the yolks in the bowl of an electric mixer fitted with a whisk attachment. In a small saucepan, combine the sugar and coconut milk. Stir to combine and then bring to a boil. As the mixture heats, begin whipping the eggs on high speed. Boil the coconut milk and sugar until the mixture reaches the soft-ball stage (238 degrees F on a candy thermometer). Remove the mixture from the heat. Stop the mixer and pour a small amount of the syrup into the egg yolks. Quickly beat on high again. Repeat twice more until all the syrup is incorporated. (You can also add the sugar syrup in a steady stream with the mixer on, but be careful not to let it hit the beater or the syrup will be flung to the sides of the bowl where it will harden.) Continue beating until the mixture is cool. Add the coconut extract. With the mixer on medium speed, begin beating in the butter 1 or 2 tablespoons at a time. When the butter is completely incorporated, scrape down the sides of the bowl and beat another 1 minutes.

TO ASSEMBLE:
Use the frosting right away to fill and frost the cooled cake or cover tightly and refrigerate until ready to use. (Bring chilled buttercream to room temperature before using, beating briefly to smooth it, if necessary.) Pat on a generous coating of the shaved toasted coconut over the sides and top and, if you like, between the layers.