Whisk together the balsamic vinegar, olive oil, salt and black pepper. In a resealable plastic bag, combine the 4 chicken breasts and ½ of the balsamic dressing. Set the remaining dressing aside for serving. Marinate the chicken in the refrigerator for 10 to 15 minutes.
With a mandoline, julienne peeler or spiral slicer, turn the zucchini into noodles. Onto a large platter, combine the zucchini noodles, chopped tomatoes and mozzarella pearls.
Heat a grill or grill pan to medium-high heat. Remove the chicken from the marinade. Discard the marinade. Grill the chicken breasts for 3 to 4 minutes per side until done. Chicken is done when it feels firm to the touch and a meat thermometer inserted in the center reaches 170°F. Remove from the grill and let rest for 2 to 3 minutes. Chop into bite-sized pieces.
Top the platter of zucchini noodles, tomatoes and mozzarella with chicken pieces. Drizzle with remaining balsamic dressing and garnish with basil chiffonade and toasted pine nuts.