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. Chill dough for one hour.
Preheat oven to 350ºF. Roll dough between sheets of parchment paper to 1/4" thickness and cut into desired shapes. If at any point the dough becomes too soft to work with, pop it back in the freezer for 10 minutes. Transfer to parchment-lined baking sheets, leaving 2" space between cookies. Bake 6-8 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.