Brown chicken well in oil in 12-inch skillet over medium heat; season with salt and pepper. Combine corn syrup, lime peel, lime juice and rum flavoring; pour over chicken. Reduce heat, cover and cook over low heat until chicken is tender, 20 to 25 minutes.
While chicken is cooking, bring broth to a boil in medium saucepan. Stir in contents of rice and seasoning packets; bring to a boil. Cover tightly; cook over low heat until all liquid is absorbed, about 25 minutes.
Remove from heat; stir in cashews. Remove chicken from skillet; keep warm. Remove excess fat from pan drippings with paper toweling. Bring pan drippings to a boil; cook, stirring constantly, until reduced to glaze consistency. Arrange rice and chicken on serving platter. Spoon glaze over chicken.