Spicy mayo and slaw:
1 garlic clove, finely grated
1/2 cup mayo
1 tablespoon Louisiana-style hot pepper sauce
1/2 small red onion, thinly sliced
1 jalapeño, thinly sliced
4 cups thinly sliced cabbage
1/2 cup bread-and-butter pickle slices, plus 1/4 cup pickle juice
2 cups all-purpose flour
1 tablespoon ground black pepper
1/2 teaspoon kosher salt plus more
1 cup buttermilk
2 8-ounces skinless, boneless chicken breasts, halved crosswise
Peanut or vegetable oil
4 white sandwich rolls
2 tablespoons butter
Spicy Mayo and Slaw
Mix garlic, mayonnaise, and hot pepper sauce in a small bowl; cover and chill. Toss onion, jalapeño, cabbage, pickles, and pickle juice in a large bowl to combine; cover and chill.
Fried Chicken Sandwich
Whisk flour, pepper, and 1/2 teaspoon salt in a shallow bowl. Pour buttermilk into another shallow bowl. Working with 1 piece at a time, dredge chicken in flour mixture, shaking off excess. Dip in buttermilk, allowing excess to drip back into bowl. Dredge again in flour mixture, shaking off excess.
Pour oil into a large heavy skillet to a depth of 1/2″. Prop deepfry thermometer in oil so bulb is submerged. Heat over medium heat until thermometer registers 350°F.
Fry chicken until golden brown and cooked through, about 3 minutes per side. Transfer to a wire rack set inside a baking sheet; season with salt. Spread cut sides of rolls with butter. Heat another large skillet over medium heat. Working in batches, cook rolls buttered side down until browned and crisp, about 1 minute.
Spread with spicy mayo. Build sandwiches with rolls, chicken, and cabbage slaw.