Make the ginger thumbprint cookie dough. Combine the flour with baking soda, cinnamon, ginger, salt, nutmeg, and cloves. Set aside.
In the bowl of a stand mixer, cream together the butter with brown sugar until light and fluffy, about 3 minutes. Add the molasses, beating until combined. Add the egg and vanilla, mixing until fully combined.
With the mixer on low, slowly add the flour mixture, mixing just until combined. Chill the cookie dough for at least 2 hours or until firm.
Meanwhile, make the pomegranate kumquat jam. Chop the kumquats into small chunks, reserving the seeds. Wrap up the seeds in a cheesecloth and place them in a medium saucepot with the chopped kumquats, pomegranate juice, sugar, star anise, cinnamon stick, and cloves. Bring to a boil over medium heat and continue to cook until thickened, about 8-10 minutes. Remove from heat and discard the cheesecloth with the seeds. Blend the jam until smooth and let cool completely.
Remove the cookie dough from the fridge. Portion the dough into individual balls about 2 tablespoons each. Roll each portion into a ball and coat them in the coarse sugar, covering all sides. Place the cookies two inches apart on a lined baking sheet. Chill the cookies for another 10 minutes.
Preheat oven to 350 degrees F.
Bake the ginger thumbprint cookies for 10-12 minutes or the bottoms of the cookies are lightly golden. Remove from the oven and carefully use the back of a small spoon to create an indentation in the center. Let cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Spoon about ½ teaspoon of the jam in the center of the cookie. Serve and enjoy!