Place chicken on a large plate and drizzle with 1 tablespoon of the olive oil. Sprinkle on the green onions, half the thyme, and salt and pepper to taste. Turn chicken to coat it well. Let it marinate at room temperature, turning it once, for 30 minutes.

Whisk vinegar, shallot, remaining thyme, and salt and pepper to taste in a small bowl. Add the 4 tablespoons of the oil in a thin stream, whisking as you go. Continue whisking until the vinaigrette has emulsified (comes together into a dressing).

Heat the grill rack, brush it with a little oil, and place chicken on it. Grill chicken 5 to 6 inches over the heat until it is cooked through on each side, about 4-5 minutes per side.

When chicken is done, let it stand 5 minutes off the heat to allow the juices to settle throughout the meat. Meanwhile, toss romaine, watercress, cabbage, and mango with just enough of the vinaigrette to coat the salad well. Transfer the salad to a platter, slice the chicken, and arrange slices over the salad. Serve the remaining vinaigrette on the side for diners to add as they please.