Whisk flour, cocoa powder, cream of tartar, baking soda and salt together.
Using a stand mixer fitted with the paddle attachment, beat butter, shortening, sugar, and brown sugar on medium speed until pale and fluffy, about 3 minutes; scraping down sides of bowl as needed. Reduce speed to low and add eggs, one at a time, and vanilla, and beat until incorporated; scraping down sides of bowl as needed.
Reduce speed to low and slowly add flour mixture until just combined. Using a spatula give dough a final stir, making sure no dry flour remains at bottom of bowl. Transfer dough to refrigerator and chill 30 minutes.
Meanwhile, heat oven to 375°F with rack set in middle position. Line two baking sheets with parchment paper.
Whisk together remaining ¼ cup sugar, five spice and cayenne.
Working with 2 tablespoons of dough at a time (about 1 ounce), roll dough into balls, then roll in sugar-spice mixture to coat; arrange 2 inches apart on prepared sheets.
Bake cookies, 1 sheet at a time, until edges of cookies are set but centers are soft, puffy and cracked (they should look underbaked and raw between the cracks), 10–12 minutes, rotating sheet halfway through baking.
Let cookies cool on sheet 10 minutes, then transfer to a wire rack to cool completely. Repeat rolling, baking and cooling with remaining dough.