Brown the butter: Melt the butter in a light-colored saucepan over medium heat, swirling occasionally. Cook until it foams, quiets, and small brown flecks appear on the bottom, about 5 to 6 minutes, then pour into a bowl to stop the cooking.
Cool the butter slightly: Let the browned butter sit for 5 to 10 minutes until just warm to the touch. This keeps it from scrambling the egg yolk in the next step.
Mix sugars and pumpkin: Whisk the browned butter with the granulated sugar, brown sugar and pumpkin puree until smooth and glossy, about 1 minute.
Add egg and vanilla: Beat in the egg yolk and vanilla extract until the mixture lightens in color slightly and looks well combined.
Whisk the dry ingredients: In a separate bowl, whisk together the flour, cream of tartar, baking soda, cinnamon, nutmeg and salt until evenly distributed.
Combine wet and dry: Fold the dry ingredients into the pumpkin mixture until no streaks of flour remain. The dough will be soft and slightly sticky.
Chill the dough: Cover the bowl and refrigerate the dough for at least 1 hour, or up to 24 hours, so it firms up enough to scoop and hold its shape while baking.
Preheat and prep pans: Preheat the oven to 350°F and line two baking sheets with parchment paper.
Make the cinnamon sugar: Stir together the 1/4 cup granulated sugar and 1 tablespoon cinnamon in a small bowl for rolling the dough balls.
Scoop and roll the dough: Scoop 1-tablespoon portions of dough, roll into balls, then roll each ball in the cinnamon sugar until fully coated.
Bake the cookies: Arrange the balls 2 inches apart on the prepared sheets and bake for 10 to 12 minutes, until the edges are set but the centers still look soft and slightly underbaked.
Cool and serve: Let the cookies cool on the baking sheet for 5 minutes, then transfer to a wire rack to finish cooling completely before storing.