Place the unsalted butter in a medium saucepan over medium heat. Melt the butter, stirring constantly. It will begin to foam and pop. Continue cooking until the popping stops and you see little brown bits at the bottom of the pan. Once it smells nutty and looks golden-amber, remove it from the heat immediately. Pour the butter (and all the brown bits) into a large heat-proof bowl and let it cool for 15 minutes.

Pumpkin puree contains a lot of moisture which can make cookies too puffy. Lay three layers of paper towels on a plate. Spread the 1/3 cup of pumpkin puree onto the towels and top with another paper towel. Press down firmly to soak up the excess liquid. You should be left with a thick, concentrated pumpkin paste.

Add the 1 cup of granulated sugar and 1/2 cup of brown sugar to the cooled brown butter. Whisk vigorously until well combined. Add the blotted pumpkin paste, egg yolk, and vanilla extract. Whisk again until the mixture is smooth and glossy.

In a separate bowl, whisk together the flour, baking soda, cream of tartar, salt, 1 tablespoon of cinnamon, and pumpkin pie spice. Gradually fold the dry ingredients into the wet mixture using a spatula. Mix until just combined; do not overmix, or the cookies will be tough.

Cover the dough and refrigerate for at least 30 minutes. This makes the dough easier to handle and prevents the cookies from spreading too much in the oven. While the dough chills, preheat your oven to 350°F (175°C) and line two large baking sheets with parchment paper.

In a small bowl, mix the remaining 1/4 cup of sugar and 1 teaspoon of cinnamon. Scoop about 1.5 tablespoons of dough and roll it into a smooth ball. Roll the ball generously in the cinnamon-sugar mixture until fully coated. Place the balls on the prepared baking sheets, spaced 2 inches apart.

Bake for 9 to 11 minutes. The edges should be set, but the centers should still look slightly soft. Do not overbake! Remove them from the oven and let them cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.