Heat oil in 12-inch nonstick skillet over medium-high heat for about 2 minutes. Sprinkle chicken with salt and pepper to taste; add to skillet and cook 5 minutes, turning once, until lightly browned. Drain off any fat; add broth, soy sauce, lime juice and ginger. Reduce heat and simmer chicken mixture for 20 minutes, until cooked through and the chicken is no longer pink in the center. With a slotted spoon, transfer chicken to a plate; cover and keep warm.
Cut each bunch of watercress in half; add to skillet along with the green onions and cilantro; cover and cook 2-3 minutes, until onions are tender-crisp.
Divide the greens and broth mixture between 4 serving plates. Place 2 chicken thighs over top of greens, and serve with rice, if desired.