Preheat oven to 350°F (175°C); line baking sheets with parchment.
Whisk together flour, baking powder, baking soda, pumpkin pie spice, and salt; set aside.
Beat butter, granulated sugar, and brown sugar until light and fluffy.
Add pumpkin puree, egg, and vanilla; mix until smooth.
Stir in dry ingredients just until combined; dough will be soft and sticky.
Scoop rounded tablespoons onto baking sheets, spacing 2 inches apart.
Bake 13–15 minutes, until edges are set and tops are puffed.
Cool on pan for 2–3 minutes, then transfer to a wire rack.
Beat cream cheese, butter, vanilla, and powdered sugar until creamy.
Frost cooled cookies and sprinkle lightly with cinnamon or pumpkin spice.