For the cookies: Combine the flour, cocoa powder, baking powder and salt in a bowl. Whisk together and set aside.
In the bowl of an electric mixer, cream the butter and brown sugar until fluffy. Add the peppermint extract and egg and beat until combined. Add the flour/cocoa mixture in 2 or 3 batches, mixing well after each addition. Stir in the chopped mints. Place the dough into a large ziptop bag and flatten slightly with your hand. Seal the bag and refrigerate for 30 minutes.
Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper or a silicone baking mat.
Remove the dough from the refrigerator and let it rest in the bag at room temperature for 5 to 10 minutes.
For rolling: Mix the flour and cocoa powder in a bowl, then tip it out onto the counter. Remove the dough from the bag and put it on the flour/cocoa mixture. Roll out the dough to a thickness of 1/3 inch. Cut with fun-shaped cookie cutters and transfer to the prepared baking sheet. You'll need to consolidate and re-roll the dough as you go.
Bake until set, about 12 minutes. Allow to cool completely on a wire rack set over a baking sheet.
For the royal icing: Beat the powdered sugar, milk and egg white with a mixer in a large bowl until thick and glossy. Add more powdered sugar if the icing seems too thin (the size of the egg can affect consistency). Just play with it until it's right. The icing should be stiff enough to "stand still" when piped and not run.
To decorate: Transfer the royal icing to a piping bag and swirl the icing onto the cookies to decorate them. Let set for 10 to 15 minutes.