Blot the pumpkin: 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) of pumpkin. Set aside.
Brown the 16 Tablespoons of butter for the cookies together with the 4 Tablespoons for the glaze. Place all 20 pieces of butter in a light-colored skillet over medium heat. (Light-colored helps you determine when the butter begins browning.) Stir or whisk the butter constantly as it melts.
Once melted, the butter will begin to foam. Keep stirring. After 5–8 minutes, the butter will begin browning you’ll notice lightly browned specks begin to form at the bottom of the pan and it will have a nutty aroma. As soon as the butter has browned, immediately remove from heat and pour it into a large heat-safe glass bowl or liquid measuring cup.
Scrape up the browned solids at the bottom of the skillet and add them as well. Divide and set aside 2 Tbsp (28g) of the brown butter to use for the glaze in step 8. The rest of the brown butter will go in the cookie dough in step 5. Allow to slightly cool while you continue.
Preheat the oven to 350°F (177°C). Line large baking sheets with parchment paper or silicone baking mats. Set aside.
In a large bowl, whisk together the 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 until combined, then whisk in the blotted pumpkin. Slowly stir the dry ingredients into the wet ingredients until combined. The cookie dough will be soft and 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 14–15 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 brown butter you reserved for the glaze a quick stir. If it’s no longer thin and liquid, warm it on the stove or in the microwave. Whisk in the confectioners’ sugar, milk, and vanilla extract until smooth. Lightly dip the top of each cookie into the glaze. Sprinkle a pinch of pumpkin pie spice on top of each glazed cookie, if desired. Return the cookies to the cooling rack and let sit until the glaze has set. Store covered tightly at room temperature for up to 2 days or in the refrigerator for up to 1 week.
