In large bowl, combine 1 cup butter and brown sugar; beat until light and fluffy. Add egg, vanilla and 2 teaspoons rum extract; blend well. Lightly spoon flour into measuring cup; level off. Stir in flour and nutmeg; mix well. Cover with plastic wrap; refrigerate 1 hour for easier handling.
Heat oven to 325°F. Shape dough into 1-inch balls; place 2 inches apart on ungreased cookie sheets. With thumb, make imprint in center of each cookie.
Bake at 325°F. for 11 to 14 minutes or until firm to the touch. Cool 1 minute; remove from cookie sheets.
In small bowl, blend all frosting ingredients until smooth, adding enough milk for desired spreading consistency. Spoon or pipe about 1/2 teaspoon frosting into center of each cookie.