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In a medium bowl, whisk together the flour, spice, salt, baking powder, and baking soda. Set aside.
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To brown the butter: In a medium saucepan or a medium skillet with high sides, melt the butter over medium heat. Once the butter has melted, continue to cook, keeping a close eye on it and stirring more frequently towards the end, until it browns, 5 to 7 minutes after it melts: The butter has browned when it smells nutty and there are visible brown bits at the bottom of the pan.
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To reduce the pumpkin: Once the butter has browned, turn the heat to low and add the pumpkin. The butter will foam and the mixture will look gloppy and broken. Stir in the pumpkin with a whisk or flexible spatula until combined, then increase the heat to medium and stir, occasionally at first and constantly towards the end, for 10 to 12 minutes, adjusting the heat as necessary to prevent scorching. The mixture is finished reducing when it’s mostly smooth, with slow-bursting bubbles; a spatula dragged through the center of the pan should leave a distinct line that closes in about 1/2 second. Transfer the pumpkin butter mixture to a liquid measuring cup: You should have about 1 1/4 cups (290g). If you have significantly more, return the mixture to the pan and continue to cook. (If you have a little more or a little less, that’s OK; proceed with the recipe.)
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Transfer the pumpkin butter mixture to a large bowl. Add the sugar and whisk vigorously until it's smooth, thick, and glossy, about 30 seconds. Whisk in the vanilla. Add the dry ingredients (from step 1) and stir with a flexible spatula until a smooth dough forms with no patches of flour. Place plastic wrap or reusable wrap directly on the dough’s surface, then transfer it to the refrigerator to chill for at least 30 minutes and up to several hours.
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To bake the pumpkin cookies: While the dough chills, preheat the oven to 350°F with racks in the upper and lower third. Line two baking sheets with parchment.
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Use a leveled jumbo cookie scoop or two level tablespoons to scoop the chilled dough onto the prepared baking sheets. For the neatest cookies, roll the dough into balls (they will feel slightly greasy). One at a time, roll the dough balls in the coarse sparkling sugar to coat. Using a cookie stamp or the bottom of a measuring cup or glass, press the balls to 1/2" thick (about 2" to 2 1/2" wide).
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Bake the pumpkin cookies, rotating once halfway through baking, for 16 to 18 minutes, until the centers are cooked through and the cookies feel firm. Let the cookies cool on the baking sheet before enjoying.
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Storage information: Store the pumpkin cookies airtight at room temperature for several days.
