In a medium mixing bowl, combine flour, salt, and nutmeg.
In a large bowl, beat butter for 1 minute until fluffy. Add sugar, and beat to combine. Add egg and vanilla, and beat until well mixed.
Add flour mixture to butter mixture in two additions, mixing until just combined after each addition.
Using a small cookie scoop (about 1 tablespoon) to portion dough into each cup of a mini whoopie pie pan (a mini muffin tin with 24 cups with be fine too). Press each dough ball slightly to flatten just a bit.
Bake for 14 to 16 minutes, or until edge are golden brown and cookies are set. Remove from oven. Cool cookie in the pan.
Once cookies have cooled, transfer them to a wire rack. Place the wire rack over a foil-lined baking sheet.
Glaze:
In a medium mixing bowl, whisk together powdered sugar and eggnog until it is a thick syrup consistency. {You want the glaze to run over the cookies, but not be too thin.}
Spoon 1/2 teaspoon glaze over the middle of each cookie and let glaze run over the cookie top and down the sides.
Sprinkle ground nutmeg over the tops of the glazed cookies.