Add cocoa to a large bowl or bowl of stand mixer. Whisk through to remove any lumps. Cream together butter and cocoa powder until well-combined. Add sugar and milk to cocoa mixture by adding 1 cup of sugar followed by about a tablespoon of milk. After each addition has been combined, turn mixer onto a high speed for about a minute. Repeat until all sugar and milk have been added.

Add vanilla extract and espresso powder and combine well. If frosting appears too dry, add more milk, a tablespoon at a time until it reaches the right consistency. If it appears to wet and does not hold its form, add more confectioner's sugar, a tablespoon at a time until it reaches the right consistency.