Gently pound chicken breasts into even thickness using the smooth side of a meat mallet, about 1/2 inch thick. Place chicken into a resealable plastic bag.
Whisk together olive oil, balsamic vinegar, garlic, Italian seasoning, and salt in a small bowl. Pour over chicken, coat evenly, and marinate for 4 to 8 hours.
Combine cherry tomatoes, basil, lemon juice, salt, and pepper in a small bowl.
Preheat an outdoor grill for medium heat; lightly oil the grate.
Remove chicken from marinade and grill with the lid closed for 4 minutes. Flip chicken and grill an additional 3 to 4 minutes. Top each breast with a slice of mozzarella cheese and 1 tablespoon of pesto, then top with cherry tomatoes. Close lid on grill and continue grilling until cheese is melted and chicken is no longer pink inside, about 3 more minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Using a spatula, carefully remove chicken to a serving platter.
