Preheat oven to 350°F. Line two baking sheets with parchment paper.
Whisk flour, pumpkin pie spice, cinnamon, baking powder, baking soda, salt, and cornstarch in a bowl.
Beat softened butter with brown and granulated sugars until light and creamy.
Mix in pumpkin puree and vanilla until smooth and combined.
Stir dry ingredients into the wet mixture just until no flour streaks remain.
Let the dough rest 10 minutes to thicken for easier scooping.
Scoop tablespoon-sized mounds onto sheets, spacing 2 inches. Gently flatten tops.
Bake 11–13 minutes until set and lightly golden at the edges.
Cool on the sheet 5 minutes, then move cookies to a rack to cool completely.
For frosting, beat cream cheese and butter until very smooth and creamy.
Gradually add powdered sugar, then beat in vanilla, milk, and a pinch of salt until fluffy.
Spread frosting over cooled cookies. Chill 15 minutes to help set, if desired.