In a large bowl, whisk together flour, cocoa, baking powder, baking soda, and salt.
Add butter and sugar to bowl of standing mixer. Beat on medium-high speed until very light and fluffy, about 3 minutes. Scrape down sides of bowl then beat in eggs, one at a time, then vanilla. With mixer speed on low, slowly add the flour mixture, and beat until combined. Roll dough between sheets of parchment paper to 1/4" thickness. Chill dough for one hour.
Preheat oven to 375ºF. Cut cough into desired shapes and pop them into the freezer for 10 minutes.
Transfer to parchment-lined baking sheets, leaving 2" space between cookies. Bake about 6 minutes, until slightly puffed and starting to crackle at edges. Cookies should still be soft and slightly underbaked. Transfer to wire racks to cool completely before icing.