Make the Cake: Preheat oven to 350 degrees F. Grease three 8-inch round cake pans, line the bottoms with rounds of parchment paper, grease the parchment, then flour the insides of the pans, tapping out excess; set aside.
In the bowl of an electric mixer (or large mixing bowl if you’re using a hand mixer), sift together the flour, sugar, cocoa powder, baking soda, baking powder and salt. In a medium bowl, whisk together the eggs, buttermilk, coffee, oil and vanilla.
Add the wet ingredients to the dry ingredients and mix for 2 minutes on medium speed. Scrape the sides and bottom of the bowl and mix for an additional 20 seconds (the batter will be very thin).
Divide the batter evenly among prepared pans. Bake for 20 minutes and rotate the pans in the oven. Continue to bake until a toothpick inserted into the center of one of the cakes comes out almost clean (with a few moist crumbs), about 12 more minutes. Cool the cakes (in the pans) on wire racks for 20 minutes, then carefully turn them out onto cooling racks to cool completely. [Make-Ahead Tip! At this point you can wrap the layers in plastic wrap and keep at room temperature for up to 1 day, or refrigerate for up to 3 days before frosting and serving. You can also wrap in aluminum foil and freeze for up to 1 month. Thaw in the refrigerator overnight before frosting and serving.]
Make the Chocolate Frosting: Place the chopped chocolate in a microwave-safe bowl and microwave in 30-second increments on 50% power, stirring after each, until the chocolate is melted and smooth. Set aside to cool to room temperature.
Meanwhile, using the whisk attachment of your stand mixer, whip the butter on medium-high speed for 5 minutes, stopping to scrape the bowl once or twice.
Reduce the speed to medium-low and gradually add the powdered sugar. Once all of the powdered sugar is incorporated, add the vanilla, mixing until incorporated. Add the melted chocolate and whip at medium-high speed until light and fluffy, about 2 minutes, scraping the bowl as needed to incorporate all of the chocolate.
Assemble the Cake: Place one of the cake layers on a serving platter or cake stand and spread 1 cup of the frosting over the top in an even layer. Place another cake layer on top of the frosting and spread with another 1 cup of the frosting. Place the last cake layer on top of the frosting upside-down, then frost the top and sides of the cake and decorate as desired. The cake can be kept, covered, at room temperature for up to 4 days. You can also freeze the cake, well-wrapped in plastic wrap and aluminum foil, for up to 1 month.