Line a medium bowl with a double layer of paper towels. Place the pumpkin puree in the bowl and press down to blot excess moisture. Repeat with new paper towels until you have about 1 cup (225g) of pumpkin. Set aside.

Brown the 16 tablespoons of butter for the cookies together with the 4 tablespoons for the glaze in a light-colored skillet over medium heat. Stir or whisk the butter constantly until melted, then continue stirring for about 5-8 minutes until browned. Pour into a large heat-safe glass bowl, scraping up browned solids. Set aside 2 tablespoons (28g) for the glaze.

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

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

In another large bowl, whisk the brown butter, granulated sugar, and brown sugar until combined. Whisk in the egg yolk and vanilla extract, then whisk in the blotted pumpkin. Slowly stir in the dry ingredients until combined; the cookie dough will be soft and sticky.

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

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

Give the reserved brown butter for the glaze a quick stir. If it’s no longer thin and liquid, warm it. Whisk in confectioners’ sugar, milk, and vanilla extract until smooth. Dip the top of each cookie into the glaze and sprinkle with pumpkin pie spice if desired. Return cookies to the cooling rack and let sit until the glaze has set.