Place chicken pieces in a single layer in a 9x13 glass dish or a resealable gallon plastic bag. Add balsamic and EVOO and massage to thoroughly distribute over the chicken. Marinate for at least 2 hours in the refrigerator. If marinating in a 9x13 glass dish, flip chicken halfway through.
Preheat oven to 350 degrees F. Remove the chicken pieces from marinade and place on a plate or cutting board. Liberally season each side of the chicken pieces with Jamaican Jerk Seasoning.
Bake chicken until internal temperature reaches 165 degrees F at the thickest part of the thigh, about 45-50 minutes.
Allow chicken to rest for 5 minutes before serving.