In a large bowl, whisk together the flour, ginger, cinnamon, nutmeg, cloves, baking soda, and salt. Set aside.
In a separate large bowl, using an electric mixer on medium speed, beat together the butter and brown sugar until no lumps remain. Add the egg, egg yolk, vanilla, and molasses, and continue beating until well combined. Turn the mixer to low and slowly beat in the flour mixture until everything is combined. Turn off the mixer and scrape down the sides of the bowl as necessary. Cover the bowl with cling wrap and refrigerate for 30 minutes or up to 24 hours.
When ready to bake, take the dough out of the fridge. Preheat the oven to 350°F (175°C) and line 2 cookie sheets with parchment paper or silicone baking mats. If the dough has been chilling for over 2 hours, allow it to sit at room temperature for about 20 minutes. Form the dough into cookie balls about 1.5 tablespoons in size. Roll in granulated or turbinado sugar. Place the dough balls 2 inches apart on the lined cookie sheets and bake for 8-10 minutes until the tops are just set. Allow to cool on the tray for 5-10 minutes before transferring to a wire rack to continue cooling.
While the cookies are cooling, make the eggnog frosting. In a medium bowl, beat the butter on medium speed until well softened. Beat in the icing sugar about 1/2 cup at a time until 2 cups have been beaten in. Add the eggnog 1 tablespoon at a time and optional rum extract. Add a little extra powdered sugar or eggnog as necessary until the desired sweetness and consistency is reached. The frosting should be smooth enough to spread.
Frost the bottom of one cookie with about 1 tablespoon of frosting, then place a second cookie on top. You’ll end up with about 16-18 sandwiches total.