Place the boiling water, chopped chocolate and butter in the top of a double boiler over medium high heat. Stir until the mixture is melted and smooth. Remove from the heat.
Preheat an oven to 350°F. Butter and flour 2 deep 8-inch cake pans, tapping out the excess flour from the pan. With an electric mixer, beat the eggs in a bowl until foamy, 15 seconds. Add the sugar and vanilla and continue to mix until creamy, 15 seconds. Add the chocolate mixture to the egg mixture and mix together. Sift the flour, baking powder, baking soda, and salt together.
Add the dry mixture to the chocolate mixture and mix until almost incorporated. Add the coffee and mix until well combined but do not overmix. Pour into the prepared pans and bake until a toothpick inserted into the center of the cake comes out clean and the cake pulls away from the sides of the pan, 25 to 30 minutes.
In the meantime for the frosting, place the butter, cream or milk, confectioner's sugar, melted chocolate and vanilla in the bowl of an electric mixer. Beat until smooth, about 1 minute.
When the cake is done, remove from the oven and cool on a cooling rack for 20 minutes. Run a knife around the edges of the pan and invert the cake onto the rack. Cool completely. Using one-half of the frosting spread it onto one of the cakes. Top with the other cake and frost the top.
To serve, cut into wedges.