Preheat the oven to 350 degrees F and place the oven rack in the center of the oven.
Butter, or spray with a non-stick vegetable spray, two 9-inch cake pans. Then line the bottoms of the parchment paper.Or use Pan Coat and Silicone Baking Pans
Whisk together the sugar, flour, cocoa powder, baking powder, baking soda, and salt in a large bowl.
Whisk together the eggs, coffee, milk, oil, and vanilla extract in a separate bowl. Add the wet ingredients to the dry ingredients and stir until combined. (The batter will be runny.)
Pour the batter into the prepared cake pans and bake for about 27-32 minutes or until a toothpick inserted into the center of the cake comes out clean.
Remove from the oven and let cool for about 10 minutes. Then remove the cakes from their pans and cool completely on a greased wire rack before frosting.
For the frosting
Melt the chocolate in a heatproof bowl placed over a saucepan of simmering water. Remove from heat and let cool to room temperature.
Beat the butter until smooth and creamy (about 1 minute).
Add the sugar and beat until it is light and fluffy (about 2 minutes).
Beat in the vanilla extract. Add the chocolate and beat on low speed until incorporated.
Increase the speed to medium-high and beat until the frosting is smooth and glossy (about 2-3 minutes)
Add the sour cream and mix until smooth and combined.