Make cake: Preheat oven to 350°F. Butter 2 9-inch round cake pans and line with parchment. Butter paper and dust pans well with cocoa powder.
Place cocoa in a small bowl. Pour in 1/2 cup boiling water; stir until smooth. With an electric mixer, beat butter until light and fluffy. Add sugar; beat until well blended. Add yolks one at a time, beating well after each addition. Beat in cocoa mixture. Stir baking soda into milk.
Sift flour, baking powder and salt onto a sheet of parchment. Add dry ingredients to butter mixture in 3 additions, alternating with milk. Blend in vanilla. In a separate bowl, beat egg whites until they form soft peaks. Gently fold into batter. Divide batter between cake pans; bake until a toothpick inserted into center comes out clean, 35 to 40 minutes. Cool in pans on a wire rack for 10 minutes, then turn out onto rack to cool.
Make frosting: Mix sugar, cocoa, salt and butter in a small pan. Stir in milk. Bring to a boil over high heat. Reduce heat to medium; simmer until dark and thickened, 5 minutes. Stir in vanilla. Transfer to a large bowl and cool.
Beat frosting with an electric mixer on high speed until thick and spreadable. Frosting should form thick swirls when beater is lifted.
Arrange one cake layer on a serving platter. Spread top evenly with about 1/2 cup frosting. Set second cake layer on top. Spread top and sides with remaining frosting.
This cake was very time-consuming to make. But it turned out good. The frosting was a flop. I boiled it for 5 minutes, and it was syrup. I let it cool, and it did thicken a bit after several hours, but it was soooooo grainy, not at all like a smooth frosting should be. I tossed it, and put a buttercream frosting on the cake. so disappointed, since I had spent so much time making this "from-scratch" cake for my husband's birthday.
I baked this cake for my friend's birthday. The cake itself wasn't sweet, which is usually good for chocolate, but it was a bit too non-sweet. I tried making the frosting but it wouldn't thicken up and become frosting. It was just syrup! I added confectioners' sugar to thicken it up and it ended up being too sweet and had a weird texture. I wouldn't make it again but it was worth a try. I moved on to a different recipe.
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