In a medium bowl, combine flour, cornstarch, baking soda, and salt. Set aside for a moment.
In a large bowl with a handheld electric mixer, beat coconut oil and sugar on medium speed until completely smooth and creamy. Beat in egg, eggnog, and vanilla until smooth. Mix in dry mixture until mixture forms a very thick cookie dough. Wrap dough in plastic wrap and chill 1-2 hours or until completely firm. (dough may be refrigerated up to 4 days or frozen up to 2 months)
Preheat oven to 350F. Line several baking sheets with parchment paper. Set aside. Once dough is firm, remove from fridge and divide into 3 portions. On a floured surface, roll dough portions out to 1/4-in thickness. Cut out dough using cookie cutters and place cut-outs 2-in apart on prepared baking sheets.
Working in batches, bake cookies at 350F 9-10 minutes or until edges of cookies begin to turn golden. Remove from oven and cool on baking sheet 5 minutes before transferring to a wire cooling rack to cool completely.
Make the icing OR frosting recipe you desire: In a small bowl, whisk powdered sugar and eggnog until smooth and creamy. Ice cooled cookies and let stand 10 minutes before serving. Enjoy!