1In large bowl, beat butter and sugar with electric mixer on medium speed until light and fluffy. Beat in egg and rum extract. On medium-low speed, beat in flour and salt until blended. Cover; refrigerate at least 1 hour.
2Cut fruitcake lengthwise in half. Shape each portion into 10 1/2-inch log on cooking parchment paper. Divide cookie dough in half. On lightly floured surface, roll each portion of dough into 11x9-inch rectangle. Place 1 fruitcake log on long edge of 1 dough rectangle; roll up jelly-roll fashion, starting from long side. Coat generously with nuts. Repeat with second portion of fruitcake and dough. Wrap tightly in plastic wrap; refrigerate 2 hours or until firm.
3Heat oven to 350°F. Unwrap dough; cut into 1/4-inch slices. On ungreased cookie sheets, place slices 3 inches apart. Bake 12 minutes or until lightly golden. Remove from cookie sheets to cooling racks.