Place butter and sugar in a large mixing bowl and beat on high speed until well combined. Add eggs, eggnog and vanilla extract, and mix well.

In a large mixing bowl, stir together the flour, baking powder, nutmeg and salt. Add the butter mixture to the flour mixture and mix until combined.

Shape dough into 2 disks and wrap each in plastic wrap. Refrigerate until firm, at least 1 hour.

Preheat oven to 325˚F. Working in batches, roll out 1 disk (leave the other in the refrigerator) on a lightly floured work surface to a 1/8 to ¼-inch thickness. Cut out desired shapes with 4-inch cookie cutters and place on ungreased baking sheet. Bake for 8 to 9 minutes or until edges are firm and the bottoms are a very light brown. Cool completely on a wire rack before decorating.

To make icing, place powdered sugar, vanilla extract and eggnog in a medium mixing bowl. Stir until well combined.

Spread each cookie with a thin layer of icing and sprinkle with colored sugar.