Make jerk rub: In a food processor combine jerk rub ingredients and process to a paste, stopping once or twice to scrape down sides of bowl with a rubber spatula. Cut several small slits in chicken and rub all over with jerk paste. Refrigerate, tightly covered, overnight. Preheat a barbecue grill and adjust heat to low. Grill chicken for about 40 minutes, turning once, until dark brown and crusty. Let cool and slice. Alternatively, bake chicken at 350 degrees F for 50 minutes, then broil 2 to 3 minutes until crisp.
Make dressing: On work surface sprinkle coarse salt over garlic and chop to a fine paste. Whisk garlic paste with lime juice and a few drops of hot sauce, to taste. Whisk in vegetable oil until emulsified. Assemble salad: In a bowl combine salad ingredients with chicken. Pour dressing over, toss to coat and arrange on a lettuce leaf lined platter.