In medium mixing bowl, cream together brown sugar, sugar, and 3/4 cup butter until smooth. Beat in vanilla and egg. Add flour and salt; stir into the creamed mixture until soft dough forms.
Roll into 1 inch balls, and place balls on ungreased cookie sheets approx 2 inches apart. Press your thumb into the center of each of the cookies to make an indention.
Bake 12 minutes in 350 degrees F preheated oven. Cool.
For the Filling:
In a small mixing bowl, combine rum, confectioners' sugar, and 1/4 cup butter.
Add rounded teaspoonfuls of filling onto cookies. Sprinkle with nutmeg.
Let stand until fully set then store in an airtight container.