Preheat oven to 350 degrees Fahrenheit. Line baking sheets with parchment paper.
Whisk flour, baking powder, baking soda, cinnamon, nutmeg, ginger, and salt together. Set aside.
Beat softened butter, granulated sugar, and brown sugar on high for 2 to 3 minutes until light and fluffy.
Add vanilla and egg and beat until combined. Add pumpkin puree and beat until smooth.
Add dry ingredients and mix on low until just combined. Dough will be very soft.
Scoop rounded portions onto prepared baking sheets, spaced 2 inches apart.
Bake 10 to 12 minutes until edges are just set. Cool on pan 5 minutes then transfer to wire rack. Cool completely.
Beat cream cheese and butter until smooth. Add vanilla. Add powdered sugar and beat until thick and fluffy.
Spread frosting generously over each cooled cookie. Dust with cinnamon if desired.