Prepare the grill for direct heat: If using a gas grill, preheat to medium-high (400 degrees) with the lid closed. If using a charcoal grill, light the charcoal or wood briquettes; when the briquettes are ready, distribute them evenly over the cooking area. For a medium-hot fire, you should be able to hold your hand about 6 inches above the coals for 4 to 6 seconds. Have ready a spray water bottle for taming any flames. Brush the grill grate.
Place the chicken breast halves between two sheets of plastic wrap, then pound each one to a thickness of about 1/4 inch (You can also ask the butcher to do that.)
Whisk together the oil, balsamic vinegar, salt and pepper in a liquid measuring cup to form an emulsified dressing. Transfer 2 tablespoons of the dressing to a bowl; use it to brush the chicken all over. (Reserve the rest of the dressing for the salad; discard any dressing left over from brushing the chicken.)
Place the chicken on the grill. Cook (uncovered) for 1 to 2 minutes per side or until just cooked through, with light char marks. Transfer to a plate.
Toss the watercress, nectarine and onion with the reserved dressing in a large bowl. Divide the chicken evenly among individual plates. Top each portion with one-quarter of the salad, and serve right away.