Preheat the oven to 350°F (175°C). Line two baking sheets with parchment paper or silicone baking mats and set aside.
In a medium bowl, whisk together the flour, baking powder, baking soda, salt, pumpkin pie spice, and ground cinnamon for about 20 seconds. Set aside.
In a large mixing bowl, beat the softened butter and granulated sugar on medium speed for 2 to 3 minutes until pale and fluffy, scraping down the sides halfway through.
Add the egg and beat until incorporated, about 30 seconds. Add the pumpkin puree and vanilla extract, and beat on low speed until smooth and uniform in color.
Add the dry ingredients to the wet ingredients in two additions, mixing on low speed after each just until no streaks of flour remain. Do not overmix.
Using a 1.5 tablespoon cookie scoop, portion the dough onto the prepared baking sheets about 2 inches apart. Flatten each mound slightly with the back of a spoon or damp fingers.
Bake one sheet at a time on the middle rack for 10 to 12 minutes, until the edges are just set and the centers look slightly underbaked. Let rest on the baking sheet for 5 minutes.
Transfer the cookies to a wire cooling rack and let them cool fully, about 20 to 25 minutes.
In a clean bowl, beat the softened cream cheese and butter until smooth, about 1 minute. Add the powdered sugar and vanilla, and beat until light and fluffy. Add milk a teaspoon at a time until spreadable.
Spread about a tablespoon of icing over the top of each fully cooled cookie. Let the icing set for about 15 minutes at room temperature before stacking or storing.