Blot the pumpkin. Spread the pumpkin puree on a few layers of paper towel, then top with more towels and gently press to remove excess moisture. This keeps the cookies from turning cakey.
Cream the butter and sugars. In a large bowl, beat softened butter with brown sugar and granulated sugar until light and fluffy, about 2–3 minutes.
Add wet ingredients. Beat in the egg yolk, blotted pumpkin puree, and vanilla until smooth.
Scrape the bowl as needed.
Whisk dry ingredients. In a separate bowl, whisk flour, cornstarch, baking powder, baking soda, salt, cinnamon, ginger, nutmeg, and cloves.
Combine. Add dry ingredients to wet and mix on low just until no dry streaks remain. The dough will be soft but not sticky.
Chill briefly. Flatten the dough into a rectangle, wrap, and chill for 30–45 minutes. This makes rolling easier and helps with clean swirls.
Prepare the filling. Stir together melted butter, brown sugar, cinnamon, and a pinch of salt.
It should be spreadable, not runny.
Roll the dough. On lightly floured parchment, roll dough into a 10x12-inch rectangle, about 1/4 inch thick. Square off edges with your hands or a bench scraper for even slices.
Add the swirl. Spread the filling evenly over the dough, leaving a 1/2-inch border on one long edge. Starting from the opposite long edge, roll into a tight log.
Seal the seam by pinching gently.
Chill the log. Wrap the log in parchment or plastic and refrigerate 1–2 hours, or freeze 25–30 minutes, until firm. Firm dough makes cleaner cuts and better spirals.
Preheat and prep. Heat oven to 350°F (175°C). Line two baking sheets with parchment.
Slice. Using a sharp knife, slice the log into 1/2-inch rounds. If a slice smashes, reshape into a circle and tighten the spiral with your fingers.
Bake. Arrange cookies 2 inches apart and bake 10–12 minutes, until set at the edges and slightly soft in the centers.
Do not overbake—they’ll finish setting as they cool.
Cool. Let cookies rest on the sheet for 5 minutes, then move to a rack to cool completely.
Glaze. Whisk powdered sugar, 1 tablespoon milk or cream, vanilla, and a pinch of salt. Add more liquid as needed for a thick drizzle. Spoon or pipe over cooled cookies.
Let set 15–20 minutes.