In a medium bowl whisk together flour, baking powder, baking soda, pumpkin pie spice, and salt.
In a large bowl cream together butter and sugars until lightly and fluffy. Add pumpkin, egg, and vanilla and mix until thoroughly.
Add dry ingredients to wet ingredients and mix until combined. Use a cookie scoop (about 1 1/2 tablespoons dough) to drop mounds onto a lightly greased cookie sheet leaving about 2-3 inches between each cookie dough mound. Use the palm of your hand to slightly flatten each mound (press down about 1/2 inch to flatten the top a little bit).
Bake at 350 degrees for 12-14 minutes. Allow to cool on the pan for about 5 minutes, then transfer to a cooling rack to cool completely.
To make the frosting, melt butter in a medium sauce pan. Add brown sugar, reduce heat to medium-low and stir in heavy cream. Continue to stir over medium-low heat for about 2 minutes until smooth. Remove from heat, stir in vanilla and pour into a large bowl. Add powdered sugar and mix in completely. Frost cooled cookies.
Store in airtight container at room temperature up to 5 days.