Drizzle olive oil in the bottom of a 4-quart or larger slow cooker. Place chicken on top of oil and sprinkle with salt and pepper.
In a small bowl, stir together the tomatoes, balsamic vinegar, garlic, bayleaf, basil, and onion.
Pour the tomato mixture over the chicken.
Cover and cook on low until chicken is no longer pink, juices run clear, and the chicken reaches 155 degrees F on an instant read thermometer, about 1 1/2 to 2 1/2 hours for breasts or 4 to 5 hours for thighs, depending upon your slow cooker. (Chicken is considered safe to eat at 165 degrees F, but the temperature will continue to rise as the chicken rests. Do not over cook or the chicken will be dry.)
Transfer the chicken to a plate, cover, and let rest 5 to 10 minutes (keep the slow cooker on so that the sauce mixture in it stays warm). To serve, uncover the chicken then use a slotted spoon to scoop the tomato-balsamic slow cooker sauce over the chicken. Top with mozzarella and fresh basil. Enjoy immediately.
To Store. Keep refrigerated in an airtight container for up to 3 days.
To Reheat. Warm gently, in a microwave-safe container, until heated through.
To Freeze. This recipe may be kept in the freezer for up to 3 months. To reheat, let thaw overnight in the fridge and then reheat in the microwave until heated through.