To flatten the chicken breasts, position each breast half, boned side up, on your work surface. Put your nonworking hand on top of the breast to keep it in place. Use your working hand to position the blade of a sharp carving knife horizontally, parallel to your work surface, with the blade placed halfway up the flesh on one long side. Carefully, while keeping the blade absolutely parallel, drive the knife into the flesh, moving the blade up and away from you and then down and toward you, leaving about ½ inch of flesh attached at the opposite end. Open the breast and lay it flat. If, after opening, you see that you didn’t go in far enough, use the blade to slice a little deeper.

Lay one opened breast half (cut side up) in the center of a sheet of waxed paper and place another sheet of paper on top. Using a smooth, heavy meat mallet, give the chicken several firm taps (not whacks) starting at the center and going out toward the edge of the meat. The aim is to thin the meat slightly and to level the outer surface. Run your hand over the top layer of paper and, if not level, give those higher areas several more soft taps with the mallet. Do this same procedure with the remaining chicken. When finished, remove the paper from each piece and stack the chicken in a 13 × 9-inch nonreactive dish.

Sprinkle the chicken pieces on both sides, lightly but evenly, with salt and black pepper. Mix the pesto with the lemon juice and pour this over the chicken. Using clean hands, rub the pesto all over both sides of each piece. Cover the dish and chill until 30 minutes before ready to cook.

Assemble the tomato topping. Use a whisk to combine the 1/3 cup of olive oil, the lemon juice, vinegar, garlic, mustard, salt, pepper, and chopped herbs in a medium-size nonreactive bowl. Add the prepared tomatoes and fold the mixture together, combining well. Cover the bowl and set it aside.

To cook the chicken using a gas grill, preheat the grill to high, covered. If using a charcoal grill, heat the coals and get the grill hot. If cooking indoors, turn on your exhaust fan and heat your stove-top grill pan(s) over high heat until very hot.

Grill the chicken, uncovered, turning once, until cooked through but not dry, 4 to 6 minutes total. Remove the chicken breasts to a serving platter and let them rest for a few minutes, loosely covered with aluminum foil.

Serve the chicken either hot or at room temperature, using a slotted spoon to top with the marinated tomatoes. Garnish each plate with a pretty basil sprig. Alternatively, the chicken and the bowl of dressed tomatoes can be passed separately at the table.