Preheat oven to 350 degrees.
In a medium bowl, mix together the flour, pumpkin pie spice, baking powder, baking soda, and salt. Set aside.
In the bowl of a stand mixer (or in a large bowl with a hand mixer), cream together the butter and granulated sugar.
Add the pumpkin, brown sugar, egg, and vanilla extract. Mix until fully combined.
Add the dry ingredients to the wet ingredients and mix well.
Drop heaping tablespoons full of cookie dough on a cookie sheet and bake 15-20 minutes, until a fork inserted in the center of a cookie comes out clean.
Allow the cookies to fully cool and drizzle the glaze on top.
To make the glaze: Combine the powdered sugar, milk, butter, and vanilla extract in a medium bowl. MIx with a fork until a glaze is formed. Use the fork to drizzle the glaze on the cooled pumpkin spice cookies.
Store uneaten glazed pumpkin spice cookies in an air tight container.