Pre-heat oven to 350 degrees and line two large baking sheets with parchment paper or a Silpat.
In a large bowl, whisk together flour, ginger, cinnamon, cloves, baking soda, and salt.
In the bowl of a stand mixer or in a large bowl with a hand mixer, cream together butter and 1 cup sugar until light and fluffy, about 2-3 minutes.
Add egg and mix until combined. Beat in molasses and eggnog.
Mix the dry ingredients into the wet ingredients until just blended, taking care not to over-mix.
Place remaining 3 Tbsp sugar in a bowl. Using your hands, form balls with about 2 Tbsp of dough at a time. Roll balls in sugar and place on prepared baking sheet about 2 inches apart. Use your hand to gently flatten the dough balls a bit.
Bake for 8-10 minutes until cookies are just turning golden around edges but are still soft. Let cool on baking sheet for a few minutes before moving to wire racks to cool completely.
When cookies are cooled, dip in eggnog icing to coat half the cookie. Top with sprinkles and let rest for about 20 minutes to allow icing to harden.