Heat oven to 350 degrees. Line cookie sheets with parchment paper, baking mats, or spray with cooking spray.
In a large mixing bowl, add softened butter, oil, pumpkin, granulated sugar, powdered sugar, vanilla and eggs. Use a hand held blender (or in the bowl of a stand mixer) and blend together until smooth and combined. It's ok if it appears lumpy (normal for pumpkin doughs).
Add the flour, pumpkin pie spice, cinnamon, nutmeg, baking soda, cream of tartar, salt, and cloves. Blend together on low speed, increase speed as need, until everything is combined and mixed well together.
Take about 2 tablespoons of dough and roll into a ball (about the size of a golf ball) place 12 dough balls per cookie sheet.
Dip the bottom of a glass cup into granulated sugar and press down on each dough ball to flatten to baout ½-inch thickness. Dip the cup between each cookie. If needed, very lightly spray the bottom of the glass cup.
Bake for 8-9 minutes. Cool the cookies on the cookie sheet for about 1-2 minutes before adding the glaze. You want them to still be warm so the glaze softens when it hits the cookie.
To make glaze combine all ingredients in a bowl and whisk together. Use more or less milk depending on how thick you want the frosting. Spread onto each cookie and let sit for several minutes to harden.