Prepare the smoker for barbecuing, bringing the temperature to 200°F to 220°F.
In a small mixing dish, combine the rub ingredients. Rub the breasts with the paste and leave them out at room temperature for about 20 minutes.
To make the sauce, whisk together the mayonnaise and half of the vinegar. Add 1 tablespoon of water and seasonings, then whisk until smooth. Spoon out ¼ cup of this mixture to use as a mop later on; set it aside for now. Note that both the sauce and mop can sit at room temperature for approximately 30 minutes before cooking or eating; however, if there is any delay beyond that window, refrigerate them as a safety measure.
Drizzle one-third of the mop over the breasts. Place the chicken in the smoker and cook until done, 25 to 30 minutes. After 15 minutes in a wood-burning pit, turn the breasts and mop again. Other smokers don't require you to turn or mop the breasts during cooking; simply drizzle them with an additional mop as soon as they come out of the smoker.
To pull the chicken, remove any large pieces of skin and discard them. Remove the chicken from the bone in chunks or strips, using your hands, two forks, or whatever you find easiest. Place the meat in a mixing bowl and drizzle with some of the reserved barbecue sauce mixtures; toss to coat. Season the pulled chicken to taste with salt and pepper.
Assemble the sandwiches, slathering sauce on one side of all the bread slices. Arrange a lettuce leaf on each of six bread slices, then top each with a chicken breast and a hearty dollop of chowchow. Top each with a remaining bread slice, halve, and serve with the remaining sauce on the side.