Preheat your oven to 350°F. Grease and line 2 9x13-inch metal baking pans with parchment paper. If using one pan, extend baking time to 30-35 minutes.
In the bowl of a stand mixer, combine melted butter, granulated sugar, vanilla extract, and eggs. Mix on medium speed until well combined.
In a separate medium bowl, sift together cocoa powder, flour, baking soda, baking powder, and salt. With the mixer on low, add half of the dry ingredients, then pour in the milk. Add the remaining dry ingredients and the coffee. Mix until fully combined.
Evenly divide the batter into the prepared baking pans and bake for 20-24 minutes or until a toothpick inserted into the center comes out clean. Allow the cakes to cool in the pans for 30 minutes at room temperature, then refrigerate until cold, about 1 hour or overnight.