Combine fresh tomatoes (cherry or grape tomatoes), basil, garlic, olive oil, vinegar, salt, and pepper in a medium bowl to make marinated tomatoes. Set aside.
Season both sides of the chicken breasts with Chicken Rub or lemon herb seasoning.
Place seasoned chicken on preheated grill grates, close the lid, and cook for 15 minutes. Flip the chicken and continue cooking until the internal temperature reaches 155°F.
Spoon 2 tablespoons of pesto over each chicken breast, then add a slice of fresh mozzarella cheese on top. Continue grilling until the internal temperature reaches 160°F.
Remove the grilled chicken from the grill and let it rest for 5 minutes. Top with marinated tomatoes and serve.