Brazilian Chicken Salad Sandwich

In a medium bowl, whisk the mayonnaise, 1 tablespoon lime juice, 1/2 teaspoon salt and 1/4 teaspoon pepper. Add the chicken, raisins, celery and onion. Season to taste with salt and pepper.
In another medium bowl, combine the oil with the remaining lime juice, 1/2 teaspoon salt and 1/4 teaspoon pepper. Mix in the carrots, beets, and cilantro.
Put a lettuce leaf on 4 of the bread slices, then top with the chicken salad, carrot mixture, another lettuce leaf, and another slice of bread.