The Hickory Dickory Cluck Chicken Sandwich

Ingredients

• 1/4 cup apple cider vinegar
• 3 tbsp prepared coarse-ground mustard
• 3 cloves garlic, peeled and minced
• 1 lime, juiced
• 1/2 lemon, juiced
• 1/2 cup brown sugar
• 1 1/2 tsp salt
• ground black pepper, to taste
• 6 tbsp olive oil
• 6 boneless, skinless chicken breast halves
• 1/2 cup mayonnaise
• 3 tbsp finely-chopped red onion
• 1 1/2 tbsp apple cider vinegar
• 1 tsp celery seed
• 1/2 tsp kosher salt
• 1/2 tsp freshly-ground black pepper
• 1/2 head of green cabbage, thinly-sliced or shredded
• 1/2 head of purple cabbage, thinly-sliced or shredded
• hot sauce, to taste
• hickory-smoked BBQ sauce
• 6 Kaiser roll