Preheat the oven to 350°F (175°C). Cover two baking sheets with parchment paper and reserve them for the cookie dough.
In a medium bowl, thoroughly whisk together the plain flour, unsweetened cocoa powder, instant espresso powder, baking soda, and baking powder. Set the combined dry ingredients aside.
Transfer the butter to a large microwave-safe bowl and melt it. Leave the melted butter to cool before continuing.
Once the butter has cooled, whisk in the packed dark brown sugar, vanilla extract, and salt until the mixture becomes smooth and no noticeable lumps remain.
Add the whole egg and extra egg yolk, mixing until they are fully incorporated into the butter and sugar mixture.
Fold the prepared dry ingredients gently into the wet mixture, stopping as soon as a cohesive dough forms. Avoid excessive mixing so the finished cookies retain a soft texture.
Portion the dough into balls using approximately 2 tablespoons of dough for each cookie. Arrange them on the parchment-lined baking sheets, leaving about 2 inches of space between each portion. Keep the dough balls rounded rather than flattening them.
Bake one tray at a time in the preheated oven. Turn the baking sheet halfway through the baking period so the cookies bake evenly.
Continue baking for approximately 15 minutes, or until the cookies appear slightly puffed and their outer edges have only just begun to set.
Remove the tray from the oven and allow the cookies to remain on the baking sheet for 5 minutes. Carefully transfer them to a wire cooling rack and leave them until completely cooled.
For the glaze, sift the powdered sugar and cocoa powder together into a medium mixing bowl.
Add 4 tablespoons of the milk along with the vanilla extract, then whisk until the glaze is smooth. Gradually incorporate additional milk as necessary, using only enough to create a glaze with a pourable consistency.
Position a wire rack over a rimmed baking sheet. Arrange the fully cooled cookies on the rack with approximately 1/2 inch of space between them.
Spoon the prepared chocolate glaze over the cookies, then leave them undisturbed until the coating has set firmly.