Preheat the oven to 350 and prepare a cookie sheet with parchment paper. Set to the side.
In a medium-sized bowl, combine the flour, baking soda, cornstarch, cinnamon, ginger, cloves, and salt. Mix to combine and set to the side.
In a medium-sized bowl with a hand mixer or in the bowl of a stand mixer with a paddle attachment, cream the butter with the sugar. Add the egg, vanilla, and eggnog and mix. Scrape down the bowl as needed.
Gradually mix the flour mixture into the butter mixture. Scrape down the bowl as needed.
Bake the cookies in the oven for 9-11 minutes. They will be soft when done. Let the cookies cool on the cookie for 10 minutes before transferring to a cooling rack.
While the cookies are baking, make the icing. Cream the butter until light and fluffy, 1-2 minutes. Add the powdered sugar, ½ cup at a time. When the icing starts to form, add in the eggnog, vanilla, and salt. Add more or less powdered sugar depending on how sweet you want the icing.
Ice the cookies when they are completly cool using an offset spatula or a spoon. Sprinkle with cinnamon when done.
