Beat butter until creamy. Beat in cocoa, mixing until combined. Stir espresso into vanilla extract then mix into the cocoa/butter mixture. Scrape down sides and bottom of bowl. Add powdered sugar, 1 cup at a time, alternating with milk. Beat each time until combined, scraping down the bowl as needed. Once all of the milk and sugar has been added, increase mixer speed and beat until fluffy. Add more sugar or milk as needed for consistency.