Line a medium bowl with a double layer of paper towels. Place the pumpkin puree in the bowl. Using another paper towel, press down to blot excess moisture from the pumpkin. You may need to repeat a couple of times with new paper towels. After blotting, you will have about 1 cup (225g) pumpkin puree. Set aside.

Brown the 16 tablespoons of butter for the cookies together with the 4 tablespoons for the glaze. Once browned, immediately remove from heat, and pour into a heat-safe glass bowl or liquid measuring cup, including all the browned solids at the bottom of the pan. Set aside 1 cup (226g/240ml) to use in the cookie dough, and reserve the rest for the glaze.

Preheat the oven to 350°F (177°C). Line baking sheets with parchment paper or silicone baking mats.

In a large bowl, whisk the oats, flour, pumpkin pie spice, cinnamon, baking soda, and salt.

In another large bowl, whisk the reserved 1 cup (226g/240ml) brown butter, the granulated sugar, and brown sugar until combined. Whisk in the egg yolk and vanilla, and then the pumpkin puree. Slowly stir the oat mixture into the pumpkin mixture. The dough should be soft and very sticky.

Scoop the cookie dough, about 2 tablespoons (45g) of dough each, and arrange them 3 inches apart on the prepared baking sheets.

Bake for 14 to 15 minutes or until the edges of the cookies are lightly browned and set. Cool the cookies on the baking sheet for 10 minutes before transferring to a cooling rack to cool completely.

Give the reserved brown butter a quick stir. If it’s no longer thin and liquid, warm it gently on the stove or in the microwave. Whisk in the powdered sugar, milk, and vanilla until smooth.

Lightly dip the top of each cookie into the glaze. Sprinkle a pinch of pumpkin pie spice (if using) on top of each glazed cookie. Return the cookies to the cooling rack and let sit until the glaze has set.

Store cookies covered tightly at room temperature for up to 2 days or in the refrigerator for up to 1 week.