Heat oven to 350°F. In large bowl, break up cookie dough. Stir or knead in remaining cookie ingredients until well blended. Shape dough into 8 (2-inch) balls, using about 1/4 cup dough per ball. Work with 1 ball of dough at a time; refrigerate remaining balls of dough until needed.
On lightly floured surface, roll dough into 12-inch rope; cut into 2-inch pieces. Place 2 inches apart on ungreased cookie sheets. Repeat with remaining dough; refrigerate until ready to bake.
Bake 7 to 11 minutes or until cookies are set and edges just begin to brown. Cool 1 minute; remove from cookie sheets. Cool completely, about 15 minutes.
Meanwhile, in small bowl, mix 1 tablespoon milk, the butter, powdered sugar, 1/4 teaspoon nutmeg and 1/4 teaspoon rum extract with spoon until creamy, adding enough milk for desired spreading consistency.
Frost cooled cookies. Sprinkle with 1/4 cup pecans; press in lightly. Let stand until frosting is set, about 15 minutes. Store between sheets of waxed paper in tightly covered container.