Brown the butter: In a light-colored medium saucepan, melt the butter over medium heat, swirling the pan periodically. It will foam, then crackle and sputter (this is the water cooking off). When the sputtering slows, watch closely and swirl the pan often. As soon as golden flecks start to gather at the bottom, whisk the butter constantly. Continue cooking and stirring until the flecks turn amber colored and the butter smells deliciously nutty. Remove from the heat.
Immediately stir the granulated sugar, brown sugar, cinnamon, pumpkin pie spice, and salt into the hot butter. Set aside to cool for 10 minutes.
While the mixture cools, blot the water out of the pumpkin. Scoop the puree onto a plate and press with a double layer of paper towel. Lift the towel away, scoop the pumpkin back into a pile, then repeat 2 to 3 more times, until most of the water has been removed and the pumpkin has a consistency like thick frosting. Measure out 1/3 cup of the blotted pumpkin. (This step concentrates the pumpkin's flavor and will keep your cookies from being cakey.)
Whisk the egg yolk and vanilla into the cooled butter mixture until smooth.
Whisk in the pumpkin, stirring until the mixture is completely smooth and has no small lumps. It will have the consistency of caramel.
In a separate bowl, whisk together the flour and baking soda. Add to the wet ingredients and fold gently just until no white bits of flour remain. The dough will be soft.
Scoop the dough into a bowl, cover with a sheet of plastic, pressing the plastic against the top, and refrigerate for at least 1 hour or up to 2 days.
When you are ready to bake, place a sheet pan in the center of your oven and preheat to 350°F. In a shallow bowl, stir together the rolling ingredients: sugar, cinnamon, and pumpkin pie spice. Line a baking sheet with parchment paper.
With a cookie scoop or spoon, portion the dough by 1 1/2-tablespoonfuls and roll into balls. Roll each ball in the sugar mixture, then place on the prepared baking sheet, leaving 2 inches between each. Return any unscooped dough to the refrigerator between batches.
Bake the cookies until they are puffed and the edges look dry and are barely turning just the tiniest bit more golden, 9 to 10 minutes. They will look underdone and will continue to set on the baking sheet.
Let the cookies rest on the baking sheet for 10 minutes (they'll sink and wrinkle), then transfer to a rack to cool completely. Repeat with the remaining dough, allowing the baking sheet to cool completely between batches, or the cookies will spread.
Notes
TO MAKE AHEAD: Refrigerate the cookie dough for up to 2 days. Scoop and roll just before baking.
TO STORE: Keep baked cookies in an airtight container at room temperature for up to 5 days.
TO FREEZE: Portion the dough into balls and freeze until firm, then transfer to an airtight freezer container. Roll in the spiced sugar before baking.