Preheat oven to 375F. Line 2 or 3 baking sheets with parchment paper or silicon baking mat, set aside. This would also be a great time to brown the butter and place it in the fridge! See first step in frosting below. 

In the bowl or a standing mixer fitted with the paddle attachment, beat butter, sugar, brown sugar and vanilla on medium speed until smooth and creamy, scraping down the sides and bottom of the bowl with a rubber spatula as needed. Add pumpkin and eggs and mix on medium speed until combined. 

In a separate bowl, whisk together flour, baking soda and pumpkin pie spice. Gradually add to wet ingredients, mixing on low speed until fully incorporated. 

Drop heaping tablespoon of dough onto prepared baking sheets. Bake for 10 – 12 minutes, until the edges are set and the top springs back when touched. Transfer to cooling rack to cool completely. 

Place the butter in a light-colored skillet or saucepan and melt over medium heat, stirring occasionally. The butter will start to foam and turn a golden brown color with a nutty aroma. You'll notice brown specks after about 5 minutes. Remove from heat and pour into a heat-proof bowl. Place in refrigerator until solid.

Remove butter and let it warm up slightly. Place in the bowl of a standing mixer (or a mixing bowl and use a hand mixer) and beat on medium-high speed until creamy. Add cream cheese and continue mixing on medium-high until combined. Add powdered sugar, vanilla extract and salt. Mix on low until powdered sugar is incorporated, then turn the speed up to high and beat for 1 – 2 minutes. Spread onto cooled cookies and enjoy!