Have all the ingredients at room temperature.
Position a rack in the bottom third of the oven and preheat to 350 F. Butter three 9-inch round cake pans and line the bottom of each pan with a round of parchment paper.
Place the chocolate and cocoa in a mixing bowl and whisk in the boiling water until smooth and the chocolate has melted. In another small bowl, whisk together the flour, baking soda and salt. Set aside.
In the bowl of an electric mixer fitted with the flat beater, beat the butter on medium speed until smooth and creamy, 1 to 2 minutes. Add the sugar and continue to beat until the mixture is light and fluffy, stopping to occasionally scrape down the sides of the bowl, about 5 minutes. Stream in the eggs a little at a time, beating until incorporated before adding more. Add the vanilla and beat until incorporated, 30 seconds. Add the buttermilk and beat until combined, about 1 minute. Scrape down the sides of the bowl.
On low speed, add the flour mixture in 3 additions, alternating with the chocolate mixture, and beginning and ending with the flour, until just incorporated. Do not over mix. Scrape the bottom and sides of bowl with a rubber spatula to thoroughly combine.
Divide batter evenly among the cake pans, smoothing the tops. Bake until a skewer inserted in the center comes out clean, 15 to 18 minutes. Cool on a wire rack 15 minutes. Run a thin knife between the cake and the pan to loosen. Invert each cake onto a cooling rack and peel off the parchment, then re-invert onto a lightly greased rack. Cool completely.
In a small heavy saucepan, melt butter and chocolate over low heat. Remove from the heat; cool for 5 minutes. In a large bowl, beat the confectioners' sugar, milk and vanilla until smooth. Gradually beat in chocolate mixture until frosting is light and fluffy. Spread between layers and over top and sides of cake