In resealable plastic bag set in shallow dish, combine 1/3 cup cilantro, chili sauce, 2 tablespoons lime juice, garlic, oil, and salt. Add chicken. Seal bag; turn to coat. Marinate, refrigerated, 1 hour.
Heat 12-inch skillet over medium-high heat. Add chicken mixture half at a time, and cook 3 minutes per batch, stirring occasionally. Return all chicken to skillet. Add sweet pepper; continue cooking 5 minutes more or until chicken is no longer pink, stirring occasionally. Using a slotted spoon transfer chicken mixture to large bowl. Cover; refrigerate 1 to 8 hours. Makes 8 pita sandwiches.
For dressing, in bowl combine mayonnaise, 1/2 cup cilantro, and 2 tablespoons lime juice. To serve, top each pita with a lettuce leaf. Spoon chicken mixture on half of each pita (if necessary, drain excess liquid from chicken). Drizzle with dressing.