In a medium bowl, combine the chicken with 1/2 cup of the apple, the onion, celery, 2 Tbs. of the mayonnaise, 1 tsp. of the mustard, the garlic, 1/2 tsp. salt, and 1/4 tsp. pepper. Gently mix with your hands and form four 1/2-inch-thick patties. Make an indentation in the center of each one with your thumb.
In a 12-inch nonstick skillet, heat the vegetable oil over medium heat until shimmering hot. Cook the burgers until golden-brown on one side, about 5 minutes. Flip, and continue cooking until the internal temperature reaches 165°F on an instantread thermometer, 5 to 9 minutes more.
Meanwhile, in a medium bowl, mix the remaining 3 Tbs. mayonnaise and 1 tsp. mustard with the lime juice. Add the remaining apple, the cabbage, and 1/8 tsp. each salt and pepper; toss to combine. Season to taste with more lime juice, salt, and pepper.
Serve the burgers in the rolls, topped with the slaw.