Preheat your oven to 350°F (175°C).
In a medium bowl, mix together the pure pumpkin puree, granulated sugar, brown sugar, softened butter, egg, and vanilla extract.
In a separate bowl, whisk together the all-purpose flour, baking powder, pumpkin pie spice, and salt.
Gradually add the dry ingredients to the wet mixture, stirring until just combined.
Drop rounded tablespoons of dough onto prepared baking sheets, spacing them about 2 inches apart.
Bake for 12-15 minutes or until the edges are lightly golden.
Allow cookies to cool on the baking sheet for 5 minutes before transferring to a wire rack.
While cookies cool, prepare the cinnamon frosting by creaming together the softened butter, powdered sugar, cinnamon, and milk.
Frost cooled cookies with the cinnamon frosting.