Preheat the oven to 350. Spray two large baking sheets with cooking spray.
In a large bowl beat together butter and 1 cup sugar, until light and fluffy. Add the egg and eggnog, mix to combine. Add the flour, salt, baking soda, and 1/2 tsp nutmeg, and mix until a soft dough forms.
In a small bowl, whisk together the ½ cup sugar and 1/2 tsp nutmeg. Scoop the dough into one inch balls and roll in the sugar mixture. Place on the prepared baking sheets.
Bake 10-12 minutes, or until lightly browned on the bottom. Let cool on pan for 5 minutes, then move to a cooling rack.
While the cookies are cooling, whisk together the powdered sugar and enough eggnog to make an icing that is easy to drizzle but not too runny. Drizzle with the icing onto cookies.