Brown the butter: Melt butter in a medium-sized skillet or saucepan over medium heat. Stir constantly as the butter sizzles and pops, scraping the bottom of the pan with a spatula to prevent burning, and stir until butter has browned (there will be lots of deep brown specks and the butter will have a toasty, nutty aroma). Immediately remove from heat and pour into a large heat-proof bowl. Allow to cool completely (until no longer warm to the touch) before proceeding.
Meanwhile, preheat oven to 350F (175C) and line baking sheet with parchment paper. Set aside.
Blot the pumpkin. Spread pumpkin onto a plate. Wad or fold paper towels and press firmly into the pumpkin to blot it, absorbing water until the pumpkin begins to stick to the paper towels and most of the excess moisture has been absorbed.
Make the cookie dough. Once butter has cooled, add sugars, blotted pumpkin, egg yolks, and vanilla extract and stir very well.
In a separate mixing bowl, whisk together flour, pumpkin spice, baking soda, baking powder and salt.
Gradually add dry ingredients to wet and stir until completely incorporated.
Add oats and stir until well distributed through the dough.
Scoop dough into 3 Tablespoon (66g) sized scoops. If desired, roll cookie dough balls between your palms to form smooth balls, this will yield more uniform cookies. Space cookies at least 2” (5cm) apart on baking sheet.
Transfer to center rack of 350F (175C) oven and bake for 12 minutes.
Allow cookies to cool on baking sheet for 5-10 minutes before transferring to cooling rack to cool completely before drizzling with glaze.
Make the glaze. Combine powdered sugar, brown sugar, 1 ½ Tablespoons of milk, corn syrup, and vanilla extract and whisk together until smooth and glaze ribbons smoothly off the whisk, holding its shape for several seconds before dissolving back into the bowl.
Drizzle glaze over cooled cookies and allow glaze to set before serving.