Preheat the oven to 350 degrees F. Line two rimmed baking sheets with parchment paper or silpat liners.
In a large bowl (or in the bowl of an electric stand mixer), cream together the powdered sugar, granulated sugar and butter with a handheld electric mixer until combined. Slowly drizzle in the coconut oil and mix.
Add the egg yolk, vanilla and pumpkin and mix until combined.
In a medium bowl, whisk together the flour, cream of tartar, salt, baking soda, cinnamon, nutmeg, ginger and cloves. Add the dry ingredients to the wet batter and mix well.
Scoop 2 tablespoons of dough at a time and roll into balls (if the dough seems sticky, refrigerate for 20-30 minutes or lightly grease your hands - I was able to roll them right after mixing). Place the balls on the cookie sheet, about 2 inches apart.
Using a drinking glass with a flat bottom, dip the glass in the 1/2 cup sugar (lightly grease the bottom of the cup the first time to help the sugar stick) and press each cookie to about 1/4-inch thick, dipping the glass in the sugar between each cookie.
Bake the cookies for 10-12 minutes.
For the icing, whisk together the powdered sugar, cinnamon and nutmeg. Slowly add the milk (or cream) and whisk vigorously until smooth. Use more or less cream depending on the consistency you want (the thinner it is, the easier it will set so the cookies can be stacked).
When the cookies are completely cooled, swirl a little icing on top of each cookie.