For marinade: In a blender or small food processor grind onion with garlic, gingerroot, lemon juice, and water to form a paste. In a bowl stir together onion paste and remaining marinade ingredients.
With a sharp paring knife make three 1/2-inch deep diagonal cuts in each chicken breast half, being careful not to cut all the way through. Rub chicken with marinade, rubbing it into cuts. In a large resealable plastic bag marinate chicken, chilled, overnight.
Preheat grill. Grill chicken on an oiled rack 5 to 6 inches over glowing coals until cooked through, about 7 minutes on a side, and transfer to a platter. Let chicken stand while preparing salad.
In a large bowl whisk together vinegar and salt and pepper to taste and add oil in a stream, whisking until emulsified. Add mesclun and toss lightly.
Slice chicken 1/4-inch thick and serve over salad.