Whisk buttermilk and hot sauce in a large bowl. Add chicken thighs, turning to coat. Cover and refrigerate for at least 2 hours or up to overnight. The longer soak yields more tender chicken.

Toss cabbage and carrot in a large bowl. In a small bowl, whisk mayonnaise, vinegar, mustard, celery seed, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Pour over vegetables and toss until evenly coated. Refrigerate until ready to use; the slaw improves as it sits.

In a shallow dish, whisk flour, garlic powder, paprika, cayenne if using, 1 teaspoon salt, and 1/2 teaspoon pepper. Remove chicken from buttermilk, letting excess drip off. Dredge each piece in the flour mixture, pressing firmly so the coating adheres. Shake off excess.

Pour oil into a large heavy pot to a depth of 2 inches. Heat over medium-high until a thermometer registers 350 degrees F. The oil should shimmer but not smoke.

Working in batches to avoid crowding, carefully lower chicken into the oil. Fry until deep golden brown and crisp, 5 to 7 minutes per side. The internal temperature should reach 165 degrees F. Transfer to a wire rack set over a baking sheet to drain. Let oil return to 350 degrees F between batches.

Split and toast the brioche buns. Place a fried chicken thigh on each bottom bun. Top with a generous mound of slaw and a few pickle chips. Cap with the top bun and serve immediately while the chicken is hot and crisp.