Heat a gas grill to high. Butterfly the chicken breasts by making a horizontal cut through the center of the thickest long side of the breast. Cut almost, but not entirely, through each breast. Open the breasts like a book. Cover with plastic wrap and pound to flatten slightly to an even thickness. Combine the pesto with 2 tablespoons of the lemon juice and rub over the chicken in a non-reactive dish.

Cut each tomato in half. Gently remove the seeds with your fingers, coarsely chop the flesh, and put it in a bowl. In a small bowl, combine the oil, vinegar, garlic, mustard, basil, remaining 1 tablespoon lemon juice, 1 teaspoon salt, and a generous amount of pepper. Pour this over the tomatoes and combine well. Remove the chicken from the pesto, shaking to remove excess marinade and sprinkle lightly with salt. Grill the chicken, turning once, until golden on both sides and cooked through, 4 to 6 minutes total. Transfer the chicken to a platter and serve hot or at room temperature. At the table, spoon the tomatoes over the chicken.