In small bowl, mix tomatoes, 2 Tbsp. basil, onion, 1 Tbsp. oil, vinegar, salt and pepper. Reserve. In another bowl, combine bread crumbs and remaining basil. Reserve.
Between sheets of plastic wrap, pound chicken breast halves to 3/8"-thickness. Coat each chicken breast in dressing, let excess drip off. Then coat in reserved bread crumb mix.
In large skillet over med-high heat, add remaining oil. Add chicken, in batches, cook, turning until no longer pink in centers. Transfer cooked chicken to platter, cover, and keep warm. After all chicken is cooked, return to skillet over med-low heat, top with cheese. Cover, heat until cheese is melted. Transfer to platter, top with tomato mix and serve.