In a medium bowl, whisk together the flour, baking soda, cream of tartar, pumpkin spice, and salt. Set aside.
Brown the butter: In a small saucepan, add 213g (15 tbsp) of salted butter and place over medium heat. As the butter melts, stir it with a rubber spatula to encourage even cooking. Once the butter has fully melted, it will alternate between rapid large bubbles and gentle small foamy bubbles. The butter will be ready once it turns a dark golden color and the fat solids have darkened and settled at the bottom of the pan. Immediately pour into a large bowl, scraping out the fat solids at the bottom, and weigh out to 170g (¾ cup).
While the butter is browning, press out the moisture in the pumpkin purée. Fold up a few paper towels and place the pumpkin in the center. Take another set of paper towels and press the pumpkin flat to soak up the moisture. Repeat this several times with fresh paper towels until very little moisture comes away and the pumpkin weighs 113g (½ cup).
To the brown butter, whisk in the sugars, then the dried pumpkin purée, egg, egg yolk, and vanilla until smooth and no more lumps of pumpkin remain.
Lastly, fold in the dry ingredients until fully combined.
Cover the bowl and chill for one hour.
When ready to bake the cookies, preheat the oven to 350 degrees F and line two large baking sheets with parchment paper.
Use a 2 tablespoon cookie scoop to portion the cookie dough. Roll each cookie dough between your hands to form a smooth ball, then roll in spiced sugar and place on the baking sheet about 2 inches apart.
Bake the pumpkin snickerdoodles for about 8 minutes or until puffed up and just sightly underbaked. Allow to cool completely on the baking sheet.