In a medium bowl, whisk together flour, cocoa powder, baking powder, baking soda, cream of tartar, and kosher salt. Set aside.
In the bowl of a stand mixer, fitted with a paddle attachment beat butter and sugar on mediums speed until smooth, about 2 minutes. Add egg and vanilla. Mix for another 2 minutes until well combined. Scrape down sides of bowl to ensure thorough mixing.
Reduce to low speed. Add flour mixture in two additions. Mix until just combined, when there are no longer any dry streaks of flour. Scrape down sides of bowl as needed.
Use a size 40 scoop (1 ½ Tbsp scoop) to portion out 16 balls of cookie dough. Working with one cookie dough ball at a time, flatten dough in your hand. Flatten into a thin disk using the palms of your hands.
Working with one cookie dough ball at a time, flatten dough in your hand. Flatten into a thin disk using the palms of your hands. Place chocolate peppermint patty in center of flattened cookie dough.
Use your fingers to gently stretch and wrap the cookie dough over the peppermint patty. Roll and pat cookie dough into your hands to ensure peppermint patty is fully encased.
Place assembled stuffed cookies on parchment lined sheet or plate. Cover with plastic wrap. Chill in the fridge for at least 2 hours, or overnight.
Preheat oven to 350°F. Place chilled stuffed cookies on parchment lined baking sheet, spacing them about 2-inches apart.
Bake in the center rack of oven for 10-12 minutes, rotating sheet midway through baking. Leave cookies on baking sheet for 3 minutes. Use a mini offset spatula or cookie spatula to transfer cookies to wire rack.