In the bowl of an electric mixer, cream together butter and sugars for a couple of minutes.
Add egg, eggnog and rum extract, beat until mixed. Scrape sides and bottom of the bowl.
Sift in flour, nutmeg, allspice, ground cloves, baking soda, baking powder and salt. Mix the ingredients on low speed until just combined. Scrape sides and bottom of the bowl and mix just a few times.
Refrigerate for about 30 minutes.
Preheat the oven to 350 and line baking sheet with parchment paper.
Scoop out cookie dough onto the parchment paper using a standard cookie scoop, placing them about 2 inches apart.
Bake for 11-13 minutes and transfer to a wire rack to cool.
In a small sauce pot, over medium heat, heat up eggnog.
Add powder sugar, rum extract, nutmeg and allspice. Whisk until all smooth and heated through. Let it cool just a little but not too much or it will start to harden.