Place oven rack 4-inches from heat. Preheat broiler to HIGH. Line a rimmed baking pan with foil. Toss tomatoes with 1 tablespoon olive oil and ½ teaspoon salt in medium bowl until coated. Spread tomatoes in prepared baking pan. Broil for 12 to 15 minutes or until lightly charred. Return tomatoes to medium bowl.

Place chicken on a cutting board; pat chicken dry with paper towels. Cover with plastic wrap. Lightly pound to ½-inch thickness using the flat side of a meat mallet. Remove plastic wrap; sprinkle both sides of chicken with remaining ½ teaspoon salt and ½ teaspoon pepper.

Heat vegetable oil in large skillet over medium-high heat. Add chicken. Cook for 6 to 8 minutes until lightly browned and chicken reaches 165°, turning once halfway through.

Add mozzarella, ¼ cup chopped basil, remaining 3 tablespoon olive oil, and remaining ¼ teaspoon pepper with tomatoes in bowl; toss to combine.

Serve tomato mixture alongside chicken on serving plates. Drizzle chicken with balsamic glaze and garnish with additional basil leaves, if desired.