Preheat oven to 375F. Line a baking sheet with parchment paper.
Mix together flour, baking soda, cinnamon, and salt in a large bowl. Set aside.
Cream (whisk) together sugars, butter, and vanilla in another large bowl (or a stand mixer) until completely blended. Beat in pumpkin puree and eggs until well combined.
Mix dry ingredients from step two into the wet ingredients from step three until well combined and smooth.
Scoop out tablespoon sized balls of dough and place them on the parchment paper lined baking sheet.
Mix together powdered sugar, vanilla extract, 3 tablespoons milk, and brown butter in a medium bowl until creamy and smooth. If mixture is too thick, add more milk one tablespoon at a time until frosting reaches desired consistency.
Store cookies in an airtight container. Cookies will keep fresh for about one week (maybe a few days longer). I like my frosting a little hard so I kept the frosted cookies in the fridge but they will also keep just fine at room temperature!