In a large bowl, whisk together the flour, ground ginger, cinnamon, nutmeg, cloves, baking soda, and salt until well combined. Set aside.
In another large bowl, beat together softened unsalted butter and packed brown sugar until no lumps remain. Add in egg, egg yolk, vanilla extract, and molasses; continue beating until well combined.
Turn mixer to low speed and gradually incorporate flour mixture into wet ingredients until completely combined. Cover with cling wrap and refrigerate for at least 30 minutes or up to 24 hours.
Preheat oven to 350°F (175°C) and line baking sheets with parchment paper. If dough has chilled more than two hours, let sit at room temperature for 20 minutes before forming cookie balls.
While cookies cool, prepare the eggnog frosting by beating softened butter until smooth. Gradually add sifted icing sugar and eggnog until desired sweetness and consistency is reached.