Preheat oven to 350 degrees. Spray one 10-inch bundt pan or two 9-inch round pans with a nonstick cooking spray and set aside.
In a large bowl, combine flour, sugar, cocoa, baking soda, baking powder and salt.
Make a well in the center and pour in the eggs, coffee, milk and oil. Mix until smooth; batter will be thin.
Pour into prepared pan. Bake in the preheated oven 35 to 40 minutes for bundt pan, 25 minutes for 9-inch round pans, or until a toothpick inserted into the center of the cake comes out clean. Allow to cool completely.
In a large bowl, mix together melted butter and cocoa. Alternately add powdered sugar and milk, beating to spreading consistency.
Add a small amount of additional milk, if needed. Stir in vanilla.