Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper and set aside.

In a large mixing bowl, whisk together the pumpkin purée, melted unsalted butter and brown sugar until smooth and evenly combined.

Add the egg and vanilla extract to the pumpkin mixture and whisk until incorporated and glossy.

In a separate medium bowl, sift together the all-purpose flour, baking soda, salt, pumpkin pie spice and cornstarch. Stir to distribute the leavening and spices evenly.

Fold the dry ingredients into the wet pumpkin mixture with a flexible spatula until just combined. Avoid overmixing — stop once there are no dry streaks and the dough holds together but remains soft. If using chopped nuts, gently fold them in now.

Pour 1/2 cup granulated sugar into a shallow bowl. Using a cookie scoop or tablespoon, portion dough into 1 1/4-inch rounds for small cookies or 2-tablespoon mounds for medium cookies. Roll each portion between damp palms to shape, then roll each ball in the granulated sugar so the surface is coated. Place the sugared dough balls 2 inches apart on the prepared baking sheets.

Bake the cookies in the preheated oven for 10–12 minutes for small cookies or 12–15 minutes for medium cookies, rotating the pans halfway through for even browning. They should have slightly domed tops with fine cracks and be set at the edges; centers will look soft but not raw.

Remove the baking sheets from the oven and let the cookies cool on the parchment-lined sheets for 5 minutes to finish setting. Transfer the cookies to a wire rack to cool completely; they will firm up slightly but remain chewy.

Once cooled, store cookies in an airtight container at room temperature for up to 4 days. To keep longer, refrigerate for up to 10 days or freeze for up to 2 months. Rewarm gently before serving if desired.