Preheat the oven to 350°F. Line a baking sheet with parchment paper and set it aside.
In a medium-sized mixing bowl, whisk together the flour, pumpkin pie spice, baking powder, baking soda, salt, and ground cinnamon. Set it aside.
Using a stand mixer, or a medium-sized mixing bowl and a handheld mixer on medium-high speed, cream together the softened butter and vegetable oil for 1 minute.
Lower the mixer speed to low, add the granulated sugar and brown sugar, and increase the mixer speed to medium-high. Mix for another 1 minute. Using a silicone spatula, scrape down the sides of the mixing bowl.
Lower the mixer speed to low, add in the pumpkin puree, increase the speed to medium and mix just until the pumpkin is completely incorporated. Scrape down the sides of the bowl.
Lower the mixer speed to low, add in the vanilla extract and the eggs, 1 at a time, mixing well after each egg.
Keeping the mixer speed on low, slowly add in the flour mixture. Mix just until all ingredients are well incorporated.
Using a 1 to 1 ½ tablespoon cookie scoop, scoop out the cookie dough onto the prepared baking sheet. Space the dough 2 inches apart.
Bake for 12 to 15 minutes. Allow the cookies to completely cool before frosting.
Cream Cheese Frosting
Using a stand mixer, or a medium-size mixing bowl and a handheld mixer on medium-high speed, cream together the softened cream cheese, clear vanilla flavoring, and butter for 1 to 1 ½ minutes. Scrape down the sides of the bowl with a silicone spatula.
Lower the mixer speed to low, and slowly add in the powdered sugar. Increase the mixer speed to medium-high and continue mixing for another 1 to 1 ½ minutes.
Lower the mixer speed to medium and slowly add the milk. Mix just until the frosting is smooth.
Once the cookies have completely cooled, spoon 1 tablespoon (or more if you wish) of frosting onto each cookie, smoothing out to the edges. If you wish, you can gently dust the tops of the cookie with the pumpkin pie spice.
These cookies can also be made larger (2 to 3 tablespoons) size.
You can make these as sandwich cookies with the frosting on the inside as well.