Preheat oven to 350°F. Line baking sheets with parchment paper.

In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl with an electric mixer, cream together the butter and sugar until light and fluffy; about 3 minutes. Add in the egg and 1 cup eggnog, and vanilla extract; mix until thoroughly combined.

In a medium bowl, sift together the flour, cinnamon, nutmeg, baking powder, baking soda and salt. Gradually add the dry ingredients to the sugar and butter mixture until they are fully incorporated.

Using a medium cookie scoop, drop dough by rounded spoonfuls onto the prepared cookie sheets. Bake in preheated oven for 10-12 minutes. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.

In a medium bowl, combine confectioners' sugar, cinnamon, and nutmeg. Stir in the remaining eggnog one tablespoon at a time until the desired consistency is reached.

Spread onto cooled cookies and let dry before serving.