In a large bowl, whisk together flour, cocoa, baking powder, and salt. Set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter, brown sugar, and ½ cup of granulated sugar on medium speed for about 3-5 minutes, or until light and fluffy.
With the mixer on low speed, add in egg, milk, and vanilla, and beat for about 30 seconds, or until all the ingredients are incorporated. Increase the mixer speed to medium-high speed and continue to mix until light and fluffy, about a minute.
With the mixer back on low speed, add the flour mixture to the butter mixture and beat until just combined, being careful not to overmix.
Remove the bowl from the stand mixer. Cover the bowl with plastic wrap and refrigerate for about 1 hour, or until the dough is easy to shape.
When ready, preheat oven to 350°F. Line 2 large baking sheets with parchment paper.
Using a 2 tablespoon scoop, drop the dough onto the prepared baking sheet.
Pour the remaining ½ cup of sugar onto a small plate or dish.
Roll each scoop of dough into a ball. Then, roll each dough ball in the sugar, making sure it is evenly coated. Place on the prepared baking sheet about 2 inches apart. Use the bottom of a glass to slightly flatten each dough ball.
Bake for 8-10 minutes, or until set.
While cookies are cooling, prepare the topping.