Preheat oven to 375.
In large bowl, cream together 3/4 cup butter, sugar, brown sugar and 1 teaspoon vanilla with mixer on medium (scrape down sides periodically).
Add eggs and pumpkin. Beat until blended.
Beat in pumpkin and eggs until well mixed.
Add flour, baking soda, pumpkin pie spice and salt. Beat on low until incorporated.
Drop about a tablespoon sized portions of dough on ungreased cookie sheets.
Bake 10-12 minutes until center of cookies are just firm.
Transfer to cooling rack to cool completely.
Place powdered sugar, vanilla and 3 tablespoons of milk in medium bowl. Set aside.
Heat butter in a small saucepan over medium heat until lightly browned, stirring constantly.
Add butter to powdered sugar mixture.
Beat on low until smooth.
If needed, slowly add last tablespoon of milk to make frosting easier to spread.