Preheat the oven to 350 degrees.
In a mixing bowl, cream butter and sugar together, mixing by hand.
Add in the eggs, eggnog, and rum extract, stirring to incorporate.
In a smaller mixing bowl, combine flour, baking soda, salt, nutmeg, cloves, and cinnamon. Whisk together well.
Mixing by hand, combine flour mixture with the butter mixture. Be sure to mix well, without over mixing.
Place balls of dough, roughly 1.5tbsp in size, on a baking sheet lined with parchment paper.
Bake for 11-13 minutes, until the bottom of the cookies are lightly browned.
Remove from oven and allow to cool on baking sheet for at least 5 minutes, then move to a cooling rack and allow to cool completely.
Once cookies are completely cooled, make the glaze by combining powdered sugar, eggnog, rum extract, and vanilla in a small bowl, stirring with a spoon. If the mixture is too thick or thin, use extra eggnog to thin it or extra powdered sugar to thicken it.
Glaze the tops of the cookies using the back of a spoon. Allow the cookies to sit at room temperature until the glaze is fully set.
Enjoy!