Preheat the oven to 350 degrees F. Line a standard-size muffin tin with paper liners.
Make the Cupcakes: In a measuring cup, whisk the espresso powder into the brewed coffee until completely dissolved. Add the milk and vanilla extract; set aside.
In a medium bowl, whisk together the flour, cocoa powder, baking powder, baking soda and salt.
With an electric mixer on medium speed, beat the butter and both sugars until light and fluffy, about 3 minutes. Add the egg and beat until combined. Reduce the mixer speed to low and add the flour mixture in three additions, alternating with two additions of the coffee mixture, mixing each until just combined. Using a rubber spatula, give the batter a final stir to ensure everything is incorporated.
Fill each muffin cup about two-thirds full with batter. Bake for 17 to 20 minutes, or until a toothpick inserted in the center of a cupcake comes out clean. Cool completely before frosting the cupcakes.
Make the Frosting: In a small bowl, whisk the espresso powder into the vanilla until dissolved; set aside.
Using the whisk attachment of a stand mixer, whip the butter on medium-high speed for 5 minutes, stopping once to scrape the sides of the bowl. Reduce 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 of the powdered sugar has been added, scrape the sides of the bowl and increase the speed to medium-high and whip until fluffy, about 1 to 2 minutes. Add the espresso and vanilla mixture and continue to mix at medium-high speed until it is completely incorporated, scraping the sides as necessary. Frost the cupcakes as desired (I used the Wilton 1M decorating tip).