For the brine: Combine the buttermilk, salt, pepper, sugar, cayenne, garlic and thyme in a gallon-size resealable plastic bag. Add the chicken pieces. Squeeze out the excess air, then seal the bag and refrigerate for at least 4 hours and up to overnight.
When you're ready to fry the chicken, remove from the refrigerator and let return to room temperature, about 30 minutes.
For the frying: Heat 2 to 3 inches peanut oil in a medium Dutch oven or large cast-iron skillet to 350 degrees F. Set racks on two rimmed baking sheets and keep one next to the stove.
Whisk together the flour, cornstarch, paprika, baking powder, garlic powder, salt, mustard and cayenne in a large bowl. Drizzle in the buttermilk in 3 or 4 additions, tossing the flour in between additions to make small clumps.
Remove the chicken from the brine one piece at a time and dredge well in the seasoned flour mixture. Set the chicken skin-side up on a rack.
When the oil reaches 350 degrees F, lightly dredge the thighs and legs again. Fry for 2 minutes, then add the wings. Continue to fry the chicken, keeping the oil around 330 degrees F, for 4 more minutes. Turn and continue to fry until both sides are crisp and deep golden and the interior of a thigh reads 165 degrees F on an instant-read thermometer, 4 to 6 minutes more (the chicken will take 10 to 12 minutes total from when you add the thighs and legs). Remove the chicken to the second rack to drain. Return the oil temperature to 350 degrees F. Lightly dredge the breast pieces again and fry as you did the thighs, turning halfway through, 10 to 12 minutes in all. Drain on the rack with the other pieces. Transfer to a serving platter and sprinkle with zaatar.