Stir together the lime juice, soy sauce, olive oil, and salt, then pour it over the chicken in a large dish, tossing the chicken to make sure it's all covered with the marinade.
Cover the dish and leave the chicken in the fridge to marinate for an hour or so (too much longer than that and the acid in the lime juice can make it mushy).
Whisk together the ingredients of the second marinade, and set aside.
Preheat your grill or broiler, and grill or broil the chicken (drained from its first marinade, which you then discard) turning it once, until it's cooked through, around 3 or 4 minutes per side.
Let the chicken rest for a minute, then slice it against the grain and toss it with the second marinade in a wide serving dish. Serve warm or at room temperature.