Reduce the speed to low and beat in the chocolate. Add the flour mixture in 3 parts alternately with buttermilk in 2 parts, starting and ending with the flour. Beat in the water until smooth.
Pour into the prepared pans. Bake until the cake pulls away from the sides and a toothpick inserted into the center comes out clean, 30 to 35 minutes.
Let the cake layers cool in the pans on wire racks for 5 minutes. Run a thin knife around the outside and turn out onto wire racks. Peel off the paper. Flip them over to completely cool right-side up on the wire racks. If not assembling right away, cover layers in plastic wrap and store at room temperature for 1 day.
Meanwhile, prepare the frosting: In the top of a double boiler or in a stainless-steel mixing bowl set over a pan of simmering water so that the bottom of the bowl does not touch the water, melt the chocolate. Set aside to cool.
In the work bowl of an electric stand mixer fitted with the paddle attachment, or in large mixing bowl with a handheld mixer, place the cooled, melted chocolate, the sugar, and the hot water. Beat on medium speed until smooth. Add the egg, butter and vanilla. Continue beating until the frosting is thick and fluffy and mounds beautifully on a spoon.
When ready to fill and frost the cake, hold a serrated knife horizontally and use a gentle sawing motion to shave the domed top off each layer. Fill and frost the cake, using a crumb coat, (see note) if you have time.
Store in a tightly covered container in the refrigerator for up to 2 days; let stand at room temperature for 30 minutes before serving.
Note: If you have time after all the layers are filled, applying a “crumb coat” to the sides and top of the cake ensures that no crumbs get mixed into the frosting and mar the cakes appearance. To apply a crumb coat, place a small amount of frosting in a dish and use an offset spatula to spread a thin coat all over the sides and top of the cake. Place in the refrigerator until firm, 20 to 30 minutes. Remove the cake from the fridge and frost the sides and top.