Preheat oven to 375 degrees F. Line two baking sheets with parchment paper.

Beat together all ingredients for the cream cheese mixture and set aside.

Make the cookie dough. Beat pumpkin, egg, sugar, oil, vanilla, cinnamon, pumpkin pie spice, salt, baking soda and baking powder together in a mixing bowl – you can do it by hand or with a mixer. Add the flour and stir just until it is mixed in.

Using a slightly rounded tablespoon, drop gobs of dough onto cookie sheets spacing about 2 inches apart. With a dampened finger, make a small well in the center of each cookie and put a spoonful of cream cheese mixture (use a rounded ½ teaspoon measure) in the center. Bake at 375º for 13 to 15 minutes or until cookies appear set.

Let cool on baking sheet for about 5 minutes, then transfer to a wire rack to cool. When completely cool, beat the cream cheese, butter, sugar and vanilla together to make cream cheese icing. Put the icing in a heavy duty freezer bag, snip off the tip, and pipe it over the cookies. Chill the cookies for about an hour just to help set the icing a bit.