Mix butter and sugar. Add eggnog and egg and stir. In a separate bowl, combine flour, baking powder, baking soda, salt, and ½ teaspoon of nutmeg. Add to the butter mixture and stir. Chill for at least one hour. Drop by teaspoonfuls onto a greased baking sheet. Bake at 350° F for 8-10 minutes. After removing the cookies from the oven, press each cookie with the bottom of a greased flat-bottom glass to flatten a bit. Cool on a rack. For the frosting, mix powdered sugar, eggnog, and syrup. Frost the cookies and sprinkle with remaining nutmeg.