Preheat oven to 350°F/175°C. Line a baking sheet with parchment paper or silicone baking mat.
In a large bowl, cream together butter and sugars with an electric mixer until combined. Mix in the oil, pumpkin, vanilla, and eggs, until combined.
In a medium bowl, whisk together the flour, baking soda, cream of tartar, salt, and pumpkin pie spice. Add the dry ingredients to the wet ingredients and stir with a spoon until well combined. The dough will be soft, but still hold shape.
Scoop about 1 1/2 tablespoons of dough at a time and roll into balls. Place on the prepared baking sheet, about 2 inches apart.
Dip the flat bottom of a glass into granulated sugar, then use it to press each cookie to between 1/4- and 1/2-inch thick. Dip bottom of glass into sugar between each to help keep the dough from sticking to it.
Bake for 8-10 minutes. The cookies should be set, but may still look as though the center is slightly under cooked. Let cool on the baking sheet for about 5 minutes, then transfer to a cooling rack.
For the icing, whisk together the powdered sugar and pumpkin pie spice in a medium bowl. Slowly add the milk and whisk until smooth. Use more or less milk depending on your desired consistency.
Spread about 1 1/2 teaspoons of glaze on each cookie. If you want the glaze to set up and harden before serving, plan to let the iced cookies sit for an hour or two.
Notes
To get an accurate measurement of all-purpose flour for this recipe, either use a kitchen scale or spoon the flour into the measuring cup, then level it off.