Heat 1 stick (8 Tablespoons or ½ cup) of the butter in a small skillet over medium heat, stirring occasionally. The butter will melt and bubble. When the butter turns golden brown and becomes fragrant, remove from heat and pour into a heat-proof bowl or measuring cup to cool. Place in the fridge or freezer to cool more quickly.
Put the 1 cup pumpkin puree into the same skillet you just browned the butter in (no need to wash it). Stir in the 1 teaspoon grated fresh ginger, 1/4 teaspoon cinnamon, and 1/8 teaspoon freshly grated nutmeg. Cook the pumpkin mixture over medium heat until it is very thick and pasty (the texture should be similar to peanut butter). Transfer to a small bowl to cool.
Mix together 3 cups all-purpose flour, 1 ½ teaspoons cream of tartar, 1 teaspoon salt, and ½ teaspoon baking soda in a medium bowl with a whisk. Set aside.
Place the remaining stick of butter into the bowl of a stand mixer or large bowl. Add the cooled browned butter, 1 cup granulated sugar and ½ cup light brown sugar sugar. Mix on medium-high speed until the mixture is uniform and fluffy, about 4-5 minutes. Add the cooked pumpkin mixture, 2 large egg yolks, and 1 teaspoon vanilla extract and mix on medium-high speed again until light and fluffy, about 3 minutes.
Add the flour mixture to the stand mixer bowl. Mix on low speed until the flour mixture is completely incorporated into the butter mixture and forms a dough, scraping down the sides of the bowl with a spatula as needed.
Stir together 1/4 cup granulated sugar and 1 Tablespoon ground cinnamon for the cinnamon sugar coating. Set aside.
Portion the dough into 1 1/2 to 2 Tablespoon portions and roll into smooth balls in the palms of your hands. Roll the balls in the cinnamon sugar mixture until coated completely. Place the dough balls in an airtight container or cover them with plastic wrap in the fridge and chill for 1 hour. Alternatively, place in the freezer for 15-20 minutes if you're in a rush.
Move oven racks to top middle and bottom middle positions and heat oven to 350°F. Prepare 2 baking sheets by lining with parchment paper or silicone baking mats. Stir together the sugar and cinnamon in a small bowl for the cinnamon sugar coating.
Place the balls about 2 inches apart on the baking sheets and flatten slightly with the palm of your hand.
Bake until the edges of the cookies have set, and cracks are visible on the tops of the cookies about 12-15 minutes, switching and rotating the baking sheets in the oven halfway through the baking time. Let the cookies cool on the baking sheets for 10 minutes, then transfer to a wire cooling rack to cool completely. Store extras in an airtight container at room temperature for 2 days, or freeze.