Add 1/2 inch of peanut oil to a cast-iron skillet and set over medium-high heat. Set up breading: flour, cornstarch, baking powder, and a pinch of salt in a shallow bowl; in another bowl, add buttermilk, hot sauce, egg, and a pinch of salt. Combine each mixture well.
Season the chicken with salt. Coat in the dry mixture, dunk in the buttermilk, then coat again with the dry mix. Once the oil reaches temperature (360° F to 375° F), carefully add the breaded chicken thighs. (Do not overcrowd.) Fry for about 4 1/2 minutes per side. Sprinkle with salt when done.
While frying, combine the cayenne, brown sugar, garlic powder, paprika, and black pepper in a heatproof bowl. When you’re done frying the chicken, add about 1/2 cup hot frying oil to the bowl with the spices and whisk until smooth. Dunk the fried chicken in the spiced oil. If it gets too thick, thin out with a little more hot oil.
Add lettuce, mayo, and pickles, and lettuce to chicken on a sandwich bun (toast bun if you wish).