Beat the butter, brown sugar, and egg on medium-high speed in a stand mixer fitted with the paddle attachment or using a hand mixer until very light and fluffy, about 5 minutes.
Scrape down the sides of the bowl, then mix in the pumpkin, 1 tablespoon of the pumpkin pie spice, and vanilla, beating on medium-high speed until combined.
Scrape down the sides of the bowl again, spoon in the flour, cornstarch, baking soda, and salt, and mix until just combined, about 1 minute.
Cover the dough and chill it in the refrigerator for at least 1 hour and up to 1 day.
Crank up the oven to 350°F (180°C) and line two rimmed baking sheets with parchment.
Stir together the granulated sugar and the remaining 1 teaspoon of pumpkin pie spice in a small bowl.
Shape the cookie dough into 1 ½-inch balls, roll them in the spiced sugar, and place them on the baking sheets, spacing them about 2 inches (5 cm) apart. Using your hand, gently flatten each ball slightly.
Bake until the edges are set and the tops are juuuust beginning to turn golden brown, 10 to 15 minutes.
Let the cookies cool on the baking sheets for 5 minutes before transferring to a wire rack to cool completely.