Preheat the oven to 350.
In a medium bowl, cream together 3/4 cup butter, white sugar, and brown sugar until smooth. Beat in egg and vanilla.
Combine flour and salt; stir into the creamed mixture by hand to form a soft dough.
Roll dough into 1 inch balls, and place balls 2 inches apart on ungreased cookie sheets.
Make an indention in the center of each cookie using your finger or thumb.
Bake for 12 minutes in preheated oven. Cool completely.
In a small bowl, mix together 1/4 cup butter, confectioners' sugar, and rum.
Spoon rounded teaspoonfuls of filling onto cookies.
Sprinkle with nutmeg. Let stand until set before storing in an airtight container.
This recipe doesn't actually use eggnog, but they have an eggnog taste. You can substitute 1/4 teaspoon rum extract and 1 tablespoon milk for the rum."
