In a small bowl, combine oil, half the lemon juice, 1 teaspoon salt, 1/4 teaspoon pepper, and Italian seasoning or oregano and whisk to combine. Transfer to a large re-sealable bag along with chicken; seal and chill, 30 minutes. Transfer chicken to a preheated grill and discard excess marinade. Grill chicken over medium-high heat until cooked through, 5 to 7 minutes per side. Meanwhile, combine tomatoes, garlic, chopped basil, and remaining lemon juice, and season with salt and pepper. While chicken is still on grill, top each breast with 1 slice mozzarella and cover grill until cheese is melted, 2 to 3 minutes. Top chicken with tomato mixture.  Sprinkle with freshly grated Parmesan and serve.