Lightly grease the inside bottoms and the sides of two 8 round baking pans with butter. Line each pan with parchment paper and place a strip of parchment paper along the sides of each cake pan, pressing gently against the butter to hold the paper in place. Set aside.
Make the Cake Layers
In a large bowl, whisk together the cake flour, baking powder, baking soda, and salt. Set aside.
In the bowl of a stand mixer or a large mixing bowl using a hand mixer, cream the butter for 1-2 minutes until light and fluffy. Add the sugar and cream together for 2-3 minutes.
Slowly add the cooled melted chocolate and beat together.
Beat in the eggs, one at a time.
Pour in the brewed coffee and vanilla and mix.
Gradually add the dry ingredients to the wet ingredients and beat together until fully incorporated.
Pour the cake batter evenly divided into the two prepared baking pans.
Bake at 350°F for 30 to 35 minutes, or until a toothpick inserted into the center of a cake comes out clean. While the cakes are baking, clean out the stand mixer bowl and paddle attachment, and make the frosting.
Make the Frosting
In the stand mixer bowl, cream the butter for 2-3 minutes, just until light and fluffy.
While the butter is creaming, In a small bowl or liquid measuring cup, whisk together 2 Tablespoons of instant coffee with the cream or milk. Pour into the stand mixer bowl and combine with the butter and sugar mixture.
Add the remaining 1 tablespoon of instant coffee and mix just until evenly incorporated throughout the batter. These add some nice coffee crunch to the frosting!
Then gradually add the powdered sugar, one cup at a time, until fully incorporated and the frosting has reached your desired level of consistency. If our frosting is too thin and runny, add more powdered sugar, in ¼ cup increments, as needed. If you will not be assembling and frosting the cake within the next few hours, store the frosting in a glass airtight container and refrigerate. Take it out of the fridge about 15-20 minutes before you are ready to use, to soften the frosting.
Assemble the cake
Once the cakes have baked, allow them to cool on a wire rack. When they are completely cool, very carefully pop them out of the pans onto a large plate or on the wire rack. They should not stick to the pan if you've greased and covered the pan with parchment paper.
If cool to the touch, wrap the cake layers in plastic wrap and refrigerate for at least two hours and up to overnight.
After that, you can even out the cake layers, then decorate with the frosting. This frosting recipe makes enough to include a layer of frosting in the middle, and frosting all the way around and on top of the cake.
Notes
Storage: wrap the frosted cake with plastic wrap and place in the refrigerator for up to four days. This cake will stay good at room temperature for a few hours, but since the frosting is buttercream, I suggest refrigerating the frosted cake if you don't intend to serve it within a couple of hours.