Preheat oven to 375 degrees.
In a large bowl, combine the 3/4 cup butter, pumpkin puree, sugars, vanilla, and eggs and mix well. Set aside.
In a medium bowl, whisk together the flour, baking soda, pumpkin spice blend, and salt.
Combine dry ingredients with butter and sugar mixture and mix thoroughly.
Drop by small rounded spoonfulls on a prepared baking sheet, and bake for 14 minutes.
Allow cookies to cool completely.
While you're waiting, heat the 1/3 cup butter over low heat, stirring frequently, until it has lightly browned. Add the powdered sugar and vanilla to a large bowl or bowl of a stand mixer and carefully pour in the browned butter. Mix well, adding more milk by the tablespoon if the icing becomes too thick.
Spread icing on top of the cooled cookies. Garnish with a light dusting of pumpkin spice or cinnamon if desired. Store in an airtight container in the refrigerator for 5-7 days.