Line a large sheet pan with parchment paper or a silicone baking mat to avoid burned cookie bottoms. Preheat oven to 350℉.
Add flour, baking soda, baking powder, cinnamon, pumpkin pie spice, and salt to a large bowl; whisk to combine and set aside.
In a medium bowl, cream together butter, granulated sugar, and brown sugar until smooth and creamy. Add 1 cup pumpkin, egg, and vanilla. Beat until mixture is creamy. Add in dry ingredients and mix until just combined. Avoid over-mixing.
Optionally, chill dough: Cover tightly and chill dough for 30 minutes up to 10 hours. Chilling is not necessary but makes dough easier to work with and intensifies flavors.
Use a 1 tablespoon measuring spoon to scoop balls of dough and place them onto the lined sheet pan. Try to smooth or round the balls, but it's okay if they're a bit bumpy (chill in the fridge for additional time if needed for easier rolling). Leave 2 inches of room for each cookie (I only bake 8–10 cookies at a time) and flatten balls of dough slightly.
Bake for 8-12 minutes (9 minutes is perfect for my oven!), watching carefully to avoid overbaking. I like these slightly underbaked and soft, but the tops shouldn’t be shiny/wet looking at all. Remove and allow cookies to stand for another 3–5 minutes on the sheet pan before removing to a wire cooling rack.
Meanwhile, make frosting. In a stand mixer fitted with the whisk attachment, beat the room-temperature (not melted) cream cheese and butter until creamy, about 2 minutes, at medium-high speed. Add powdered sugar, salt, and vanilla. Beat until soft and creamy. Avoid over-beating or you’ll get whipped frosting.
Generously frost cooled cookies and if desired, sprinkle them with a little pumpkin pie spice.
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Notes
Storage: Do not store the cookies in a sealed plastic container. Instead, store on plates loosely covered with foil in the fridge. They’re best eaten the same day or within 2–3 days. After that, they soften too much. Freeze unbaked cookie dough balls for up to 3 months; bake straight from freezer with an additional 1–3 minutes.