You may need to seek out a specialty cheese store to locate the bocconcini. These small balls of fresh mozzarella cheese become soft and creamy when tossed with the hot pasta, chicken, and vegetables.
Lightly coat grill rack or grill pan with nonstick cooking spray. Heat grill or grill pan to medium high. Cook pasta following package directions. Season chicken with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. Grill chicken, about 6 minutes per side until cooked through. Set aside to cool slightly, then cut into thin strips. When pasta is done, drain and toss in a bowl with 2 tablespoons of the oil. Set aside.
Heat remaining 2 tablespoons oil in a 12-inch skillet over medium heat. Add garlic; cook 1 minute. Add tomatoes with their juice, zucchini and yellow pepper; cook 3 more minutes. Add vinegar, tomato paste, remaining 1 teaspoon salt and 1/4 teaspoon pepper; cook 3 minutes. Stir in chicken and heat through. Remove from heat. Toss sauce with pasta, basil and mozzarella and serve.