Preheat the oven to 350°F. In the bowl of a stand mixer fitted with the paddle attachment mix the sugar, shortening, butter, and canned pumpkin puree. Mix for 1 minute. Add the egg and vanilla extract. Mix.
In a separate bowl combine the flour, pumpkin pie spice, baking soda, cinnamon, baking powder, salt, nutmeg, and cloves. Whisk to combine. Add the dry ingredients to the wet ingredients. Mix until well combined.
Place a silicone baking mat on a baking sheet. Scoop 2 tablespoon sized dough balls onto the baking mat. Place in the oven and bake for 15-17 minutes. The cookies will turn a light golden color on the edges. Remove from the oven and let cookies cool on the pan for 5 minutes. Remove cookies to a cooling rack to cool completely.
Glaze
In a small saucepan over medium heat, melt the butter. Add milk. Bring to a boil. Remove from heat and whisk in the vanilla and powdered sugar. Drizzle the glaze over the cooled cookies.
Notes
If you want to make smaller sized cookies, you can use a 1.5 T cookie scoop and bake the cookies for 12-15 minutes.