Preheat a grill for indirect medium heat. Using a food processor, puree the 3 whole cloves garlic, the lemon peel and 1 teaspoon each lemon juice, olive oil and salt. Using your fingers, gently loosen the skin of the chicken breasts and spread the lemon mixture underneath. Drizzle 1 tablespoon olive oil on the undersides of the chicken. Season with salt and pepper.
Arrange the chicken, skin side down, on an oiled grate over direct heat, cover and grill until golden and marks appear, 3 to 4 minutes. Flip and place the chicken, skin side up, over indirect heat, cover and grill until cooked through, 15 to 20 minutes. Transfer to a work surface, drizzle with some lemon juice and let rest for 10 minutes.
Meanwhile, in a salad bowl, whisk together the finely chopped garlic, balsamic vinegar and remaining 2 teaspoons olive oil; season with salt and pepper. Add the tomatoes, cheese, onion and mesclun greens and toss; season with salt and pepper. Serve with the chicken.