Whisk flour, cream of tartar, baking soda, and salt in a medium bowl. Using an electric mixer on medium speed, beat butter, brown sugar, and 1 cup granulated sugar (and cinnamon) in a large bowl, scraping down sides, until incorporated. Scrape in seeds from vanilla bean; reserve pod for another use.
Add eggs and continue to beat until mixture is pale yellow, begins to thicken, and falls from beaters in a ribbon-like stream, about 3 minutes. Reduce mixer speed to low. Gradually add dry ingredients and continue to beat, scraping down sides and bottom of bowl, until just combined. Let sit at room temperature at least 30 minutes to hydrate flour; dough will be very loose but will thicken as it sits.
Preheat to 400°
Using a 1-oz. ice cream scoop or a generous, heaping tablespoonful, portion cookies and roll into small balls with your hands. Transfer to prepared sheet.
Bake cookies 15 minutes