Preheat grill to 350 to 400 (medium-high) heat. Cook balsamic vinegar in a 2-qt. saucepan over medium-high heat 4 to 5 minutes or until slightly thickened; remove from heat, and whisk in butter.
Sprinkle desired amounts of kosher salt and freshly ground black pepper over tomato slices and chicken breasts. Grill chicken, covered with grill lid, 6 to 8 minutes on each side or until done.
Brush chicken with 2 Tbsp. balsamic mixture; top each with 2 fresh basil leaves and 2 fresh mozzarella cheese slices. Cover with grill lid, and grill 3 minutes or just until cheese melts.
Remove from grill; top each chicken breast with 1 tomato slice; drizzle with remaining balsamic sauce. Top with torn basil.
