In a heavy stainless steel skillet, melt the butter over medium heat. It will froth, crackle, and sputter, which is perfectly normal. Stay close and stir it regularly to prevent scorching. Once the milk solids on the bottom turn golden brown and the butter smells nutty and toasty, take the pan off the stovetop.

Pour the browned butter into a glass liquid measuring cup, leaving any sediment behind; you should have just under 1 cup (184 grams). Let it cool in the refrigerator, stirring every 20 minutes, until an instant-read thermometer shows 70 to 75°F. The butter should feel cool and still be liquid.

Position the oven racks in the upper and lower thirds and preheat the oven to 350°F. Line two baking sheets with parchment paper.

Spread the pumpkin puree on a dinner plate in a thin layer. Press a paper towel gently into it to absorb excess moisture. Scrape the puree into a mound, spread it thin again, and blot with a fresh paper towel. Repeat until the puree looks like soft playdough and measures about 1/3 cup (68 to 75 grams).

Add the cooled browned butter to a large mixing bowl. Whisk in the granulated sugar and brown sugar for 1 full minute; the mixture will look like wet sand.

Whisk in the egg yolks, vanilla, and the dried pumpkin puree until the mixture is smooth.

Fold in the flour, pumpkin pie spice, fine sea salt, cream of tartar, and baking soda with a spatula until just combined and no dry spots remain. Refrigerate the dough for 5 minutes to firm it up; chill longer if the dough is very soft.

In a small shallow bowl, whisk the cinnamon and the remaining 1/3 cup granulated sugar together. Using a 3-tablespoon scoop, portion out the dough and roll each ball in the cinnamon sugar. Arrange the balls 2 to 3 inches apart on the prepared baking sheets.

Bake the cookies in batches, one sheet at a time, for 10 to 12 minutes, until the edges are golden brown and the centers are puffy and just barely set. Let the baking sheet sit on a wire rack until completely cool.

Store any leftover cookies in an airtight container at room temperature for 2 to 3 days. To freeze, form the dough into balls and freeze them solid, then transfer to a freezer bag. Before baking frozen dough balls, let them come to room temperature, roll in cinnamon sugar if not already coated, and proceed with the recipe.