Pound thighs evenly to 1/2-inch thickness; soak in buttermilk and hot sauce 20 minutes.
Whisk flour and salt. Lift thighs from buttermilk; firmly press into flour.
Heat oil in a deep, heavy pot to 350°F; keep pot no more than half full.
Fry 2 thighs at a time 6 to 8 minutes, turning once and maintaining 340°F to 350°F, until crisp and centers reach 165°F.
Drain on a rack; carefully measure 1/3 cup hot oil into a heatproof bowl.
Whisk cayenne, brown sugar, paprika, and garlic powder into measured oil; brush both sides of each thigh.
Fill each bun with one hot thigh and 4 pickle chips. Close; serve immediately.