Line a cookie sheet with parchment and preheat your oven to 350 degrees Fahrenheit.
In a large mixing bowl, beat together the 3/4 cup butter, granulated sugar, and light brown sugar, until smooth.
Add in the pumpkin puree, egg, and 1 tsp. vanilla and mix until combined.
In another bowl, whisk together the all purpose flour, baking powder, baking soda, pumpkin pie spice, and salt.
Add the dry ingredients to the wet, mixing well.
Scoop heaping scoops of the dough with a cookie scoop, and put them onto the parchment with plenty of space between.
Bake for 13-15 min, or 1-2 minutes longer for extra large cookies.
Allow the cookies to cool completely.
For the frosting, beat together the cream cheese, 3 tbsp. butter, and the powdered sugar, until combined.
Add in the vanilla, and mix until combined.
Spread some frosting onto each pumpkin cookie, and sprinkle with an extra sprinkle of pumpkin pie spice, if desired.