Preheat oven to 350°F and line two cookie sheets with parchment paper. In a small bowl, whisk together the flour, cornstarch, salt, nutmeg, and cinnamon. Set aside.
In a medium bowl, cream together the butter and sugar until light and fluffy. Add the vanilla extract and eggnog just until combined.
Add half of the flour mixture to the butter mixture until combined. Repeat with remaining flour. The dough may be crumbly, but will come together as you continue mixing. Use your hands to press the dough together, if necessary. The dough will be sticky.
Form the dough into two discs, and refrigerate until the dough is no longer sticky but doesn't crack when pressed, about 30-45 minutes.
If you have cookie stamps: Roll the dough into 2 Tbs. balls. Dust the cookie stamps in flour, and press firmly into each dough ball until the cookie is even and the design is clearly imprinted. Clean rough edges with a cookie cutter, if desired. Refrigerate the stamped cookies for 15-30 minutes.
If you do not have cookie stamps: Roll the dough on a lightly floured surface until it is 1/4'' thick. Cut out 2'' circles with a cookie cutter. Refrigerate the cookies for 15-30 minutes.
Bake the chilled cookies for 12-14 minutes, turning the pan halfway, until the tops of the cookies are set and the bottoms are golden brown. Let the cookies cool on the pan for about 5 minutes, and then transfer to cooling racks to cool completely.
In a small bowl, whisk together all of the ingredients until smooth.
Dip the top of the cooled cookies into the glaze. Set on a cooling rack and let harden. Serve or store in an airtight container for up to one week!
Chill the dough before and after stamping the cookies to make sure they do not puff up during baking.If you do not use stamps, add cute sprinkles to the glaze for decoration!