In the bowl of your stand mixer, use the paddle attachment to cream together the butter and brown sugar until light and fluffy. Then beat in the egg until incorporated. Fold in the cooled chocolate.
In a separate bowl, sift together the flour, cocoa powder, instant espresso, baking powder, and salt. With the mixer on low speed, add the dry ingredients to the wet ingredients (the dough will be a little sticky), and then add the milk and mix until combined. Turn the dough out onto your work surface. Using your hands or a rolling pin, flatten the dough into a disk. Wrap it tightly in plastic wrap and refrigerate until the dough is firm, about 45 minutes.
Preheat your oven to 350°F (180°C). Line two baking sheets with parchment paper. Pour some icing sugar onto a plate.
Shape the dough into 1-inch (2.5 cm) balls (approx.). Roll each ball in the icing sugar several times—you want them to be completely covered in sugar so that no dough shows through—and put it on a baking sheet, spacing each about 2 inches (5 cm) apart. Don’t crowd them.
Bake for 12 to 15 minutes, until the cookies have spread and cracked. The dark chocolate will show through the white sugar coating, which gives them a nice finish.
Transfer the cookies to a wire rack to cool. Store in an airtight container for up to a week or freeze them for even longer.