Preheat oven to 350°F and line baking sheets with parchment paper.
In a medium bowl, whisk together flour, pumpkin pie spice, baking powder, baking soda, cinnamon, and salt.
In a large bowl, cream butter and sugar together until light and fluffy, about 2-3 minutes.
Beat in pumpkin puree, egg, and vanilla extract until well combined.
Gradually add dry ingredients to wet ingredients, mixing on low speed until just combined.
Drop rounded tablespoons of dough onto prepared baking sheets, spacing 2 inches apart.
Bake for 13-15 minutes until cookies spring back when lightly touched. Cool completely on wire rack.
For frosting, beat cream cheese and butter until smooth. Gradually add powdered sugar and vanilla, beating until creamy.
Spread or pipe frosting onto cooled cookies and dust lightly with cinnamon.
Let frosting set for 10 minutes before serving or storing in an airtight container.