Preheat the oven to 350°F (175°C) and line two baking sheets with parchment paper.
In a medium bowl, whisk together the flour, cocoa powder, baking soda, salt, and espresso powder. Set aside.
In a large bowl, beat the butter and granulated sugar until light and fluffy, about 2-3 minutes.
Add the egg and vanilla extract to the butter mixture and beat until fully combined.
In a separate measuring cup, combine the milk and brewed coffee.
Add the dry ingredients to the wet ingredients in two additions, alternating with the milk-coffee mixture. Mix just until combined, being careful not to overmix.
Scoop the batter onto the prepared baking sheets using about 2 tablespoons per scoop, spacing them 2 inches apart.
Bake for 10-12 minutes, or until the cakes are set and spring back lightly when touched.
Remove from the oven and let the cakes cool completely on a wire rack before filling.
To make the filling, dissolve the espresso powder in the hot milk or water.
Beat the butter until smooth and creamy, then gradually add the powdered sugar.
Add the dissolved espresso mixture and vanilla extract. Beat until light and fluffy. Add additional milk as needed to reach a smooth, spreadable consistency.
Spread or pipe the filling onto the flat side of half the cooled cakes, then sandwich with the remaining cakes.