Combine the ingredients for the chicken marinade in a baking dish. Place the chicken in the marinade. Cover and refrigerate for at least 30 minutes.
Meanwhile, chop the vegetables for the salad. Mix the hot water, vinegar, red pepper and salt in a medium bowl. Stir to dissolve the sugar. Then place the cucumbers and radishes in the mixture to pickle. Toss and set aside.
Preheat the grill to medium heat. Grill the chicken for 5 minutes per side, then cover with foil and let it rest for 5 minutes before slicing.
Place the romaine lettuce on dinner plates, topped with carrots, pickled cucumbers and radishes, sliced chicken, cilantro and green onions.
Mix 2 tablespoons of the pickling liquid with mayo and olive oil. Whisk well and salt and pepper to taste. Drizzle over the top of the salads.