1Mix the espresso powder into the brewed coffee until dissolved; set aside to cool to room temperature.
2Preheat the oven to 350 degrees and line a standard sized muffin tin with paper liners.
3Whisk together the flour, cocoa powder, baking powder, baking soda, and salt.
4Beat the butter and both sugars together on medium-high speed until light and fluffy. Add the egg and beat until combined. In a measuring cup, combine the milk, brewed coffee mixture and vanilla.
5Slowly add the flour mixture, alternating with the coffee mixture, and ending with the flour mixture.
6Divide the batter evenly between the 12 liners. Bake for 17-20 minutes or until a toothpick inserted in the center of a cupcake comes out clean. Allow the cupcakes to cool completely before frosting them.
7Mix the espresso powder into the vanilla until dissolved, set aside.
8Using the whisk attachment of a stand mixer (hand mixer is also fine), whip the butter on medium-high speed for 5 minutes, stopping once to scrape the sides of the bowl.
9Reduce the mixer speed to low and add the powdered sugar a little at a time, waiting until it is mostly incorporated before adding more. Once all the powdered sugar is added, scrape the sides of the bowl and increase the mixer speed back to medium-high and whip until fluffy.
10Add the espresso and vanilla mixture and continue to mix at medium-high until it is completely incorporated, scraping the sides as necessary.