Preheat the oven to 350°, adjust oven rack to the middle position and line a baking sheet with parchment paper.
In a small bowl, whisk together all of the following ingredients: flour, baking powder, baking soda, pumpkin pie spice and salt.
In a large bowl, beat the butter and sugar at medium speed using a stand mixer or handheld electric mixer, until light and fluffy, 2-3 minutes. Beat in the egg and vanilla and pumpkin puree.
Adjust the mixer to low speed and mix in the dry mixture a little at a time until just blended together. Scrape the sides of the bowl with a spatula and finish fold in any remaining flour or wet mixture until completely combined.
Using a tablespoon or small dough scoop, drop 9 (2-tablespoon) scoops of dough onto the prepared baking sheet, spaced evenly.
Wet the back of your scoop or your fingers and lightly press the tops of each dough dough scoop until smooth and slightly flattened (about 2-3 inches round and 1 inch high).
Bake until the edges of the cookies are firm, around 15 minutes. Continue to bake 9 at a time until the dough is gone. Allow to completely cool before glazing (about 30 minutes).
FOR THE ICING:
In a small bowl, whisk all glaze ingredients together until smooth. Add additional powdered sugar to thicken if needed. The icing should be thick, but still stirrable. Dip the cookie into the icing and swirl a bit. Transfer the iced cookie to a metal drying rack with parchment or wax paper underneath. Allow the glaze to dry for about 10 minutes.