Place garlic, basil, and vinegar in a blender. With the motor running, slowing drizzle 1/3 cup plus 2 tablespoons oil into blender until fully incorporated. Season with salt and pepper and set aside.
Brush 1 tablespoon oil over each chicken breast and season with salt and pepper. Place a grill pan over medium-high heat and brush with remaining tablespoon oil. Sear each breast 2 minutes on each side. Lower heat to medium, cover, and continue to cook until chicken has cooked through, 8 to 10 minutes.
Transfer chicken to a cutting board and let rest, 5 minutes. Slice chicken on a bias into thin, long strips.
Pour romaine into a bowl and top with mozzarella balls, halved cherry tomatoes, and chicken strips. Drizzle with vinaigrette and finish with fresh basil leaves and grated Parmesan.