1. Preheat broiler. Combine first 9 ingredients in a shallow dish; let stand 15 minutes.
2. Thread 4 onion pieces and 3 chicken pieces alternately onto each of 8 (12-inch) skewers. Place skewers on broiler pan coated with cooking spray. Broil 12 minutes or until chicken is done, turning occasionally.
3. Combine nectarines and next 6 ingredients (through 1/4 teaspoon salt) in a bowl. Gently stir in avocado.