Preheat oven to 350°F (175°C). Grease and line two 9-inch round cake pans with parchment paper.
In a large bowl, whisk together flour, sugar, cocoa powder, baking soda, baking powder, and salt.
In a separate bowl, beat eggs, then add cooled coffee, milk, oil, and vanilla.
Gradually mix the wet ingredients into the dry ingredients until just combined. Do not overmix.
Divide batter evenly between prepared pans and bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean.
Allow cakes to cool in pans for 10 minutes, then remove and cool completely on wire racks.
For the frosting: Beat softened butter until creamy (about 2 minutes). Gradually add powdered sugar and cocoa powder, alternating with coffee. Add vanilla extract and salt. Beat until smooth and creamy (3-4 minutes).
Place one cake layer on serving plate and spread about 1 cup of frosting evenly over the top. Place second layer on top and cover entire cake with remaining frosting.