Preheat the oven to 350℉ and set a rack in the middle of the oven. Line two baking trays with parchment.
In the bowl of a stand mixer, combine the butter, sugar, espresso powder, baking powder, baking soda, salt, and vanilla. Beat until light and fluffy.
Add the egg to the butter mixture and beat until smooth. Add the cocoa powder and mix until combined.
Add a third of the flour to the mixture and a third of the milk to the mixture and mix until combined. Repeat twice more with another third of the flour and milk.
Scrape the sides of the bowl to ensure all ingredients have been fully incorporated and mixed.
Using a medium cookie scooper, drop a scoop of batter the size of a ping pong ball onto the lined baking tray. Leave at least 2" to allow for spreading of the batter.
Bake for 12 minutes. The tops of the cookies should be set and firm to the touch.
Remove the trays from the oven and let the cakes cool for 5-10 minutes in the tray. Using a spatula, remove the cakes from the tray and let fully cool on a wire rack.