Pound each chicken breast out to 1/4-inch thickness between 2 pieces of parchment paper. Season with salt and pepper.
Using a clean tea towel, oil the grill grates.
Grill the chicken on each side until golden and cooked through, for 3 to 4 minutes, until golden and cooked through.
While the chicken is grilling, add the hot sauce, butter, paprika, apple cider vinegar, and garlic to a medium saucepan. Place the pan on the grill grates and stir to melt all together. Dip the chicken into the saucepan to coat and then grill for another minute on each side. Remove the chicken from the grill and place on a platter. Top each cutlet with some Bblue Ccheese Sslaw.
Blue Cheese Slaw:
Using a large-holed grater attachment for a food processor, grate the cabbage, shallot, and carrot. Add to a large serving bowl.
Mix together the blue cheese, buttermilk, sour cream, mayonnaise, apple cider vinegar, celery seed, salt, and pepper together in a small small bowl. Fold into the grated veggies and season with salt and pepper, to taste. Cover with plastic wrap and let sit in the fridge for at least 1 hour to absorb flavors. Can be made a 1 day in advance.