1. Toss chicken with salt, pepper, and lime juice. Heat oil in large nonstick skillet over medium-high heat 1 minute. Add chicken; cover skillet and cook until golden brown, 6 to 7 minutes. Turn chicken, cover and cook 6 to 7 minutes more, or until instant-read thermometer inserted in thickest portion of each breast registers 170 degrees F. Wrap chicken in a large piece of foil and seal. Cool 20 minutes, then dice.
2. Meanwhile, cook bacon in same skillet over medium heat until crisp, 7 to 8 minutes. Drain.
3. For dressing, combine mayonnaise, anchovies, vinegar, parsley, chives, water, lime juice, and sugar in bowl.
4. Arrange lettuce on a large platter, then arrange chicken, eggs, avocado, cheese, and bacon in rows over top. Serve with dressing. Makes 4 servings.