Heat gas or charcoal grill. In small bowl, beat raspberry spreadable fruit, gingerroot, vinegar and soy sauce with wire whisk until well blended; set aside.
In 10-inch nonstick skillet, heat oil over high heat 1 minute. Add onion; cook and stir 2 minutes. Reduce heat to medium; cook 2 minutes longer or until onion is tender and rich dark brown. Reduce heat to low; stir in raspberry mixture. Cook 1 minute, stirring constantly. Remove from heat; set aside.
Place chicken, bone side up, on grill over medium-high heat. Cover grill; cook 15 to 17 minutes, turning frequently, until juice of chicken is no longer pink when centers of the thickest part is cut to bone (170°F). Serve chicken with onion-berry spread.