For the chicken: Place the chicken skin in a single layer in a large nonstick skillet. Cook the skin over low heat, flipping once, until it crisps up, about 10 minutes. Meanwhile, shred the chicken meat into bite-size pieces into a large bowl. Reserve the wings to nibble on or for another use.
In a large bowl, whisk together the mayo, vinegar, horseradish, mustard, pepper and salt to taste. Add the dressing to the shredded chicken and toss to coat. Set aside to let the dressing soak into the chicken.
For the pickle slaw: In a medium bowl, add the cabbage, celery salt, sugar, hot sauce, if using, pickles and pickle brine, and carrot. Toss together to mix well. Taste and the adjust seasonings and set aside.
Preheat a griddle pan over medium-low heat. Butter one side of each slice of bread. Cook the buttered sides on the griddle until toasted and golden, about 4 minutes.
Top the unbuttered sides of 4 slices of the toast with the crispy chicken skin, then top with the chicken and pickle slaw. Top each sandwich with a slice of bread with the buttered-side up. Serve.