Place the buttermilk in a resealable plastic bag, add 1 1/2 teaspoons of the salt, 1/4 teaspoon of the thyme, 1/2 teaspoon of the black pepper, and 1/8 teaspoon of the cayenne pepper and stir to combine. Add the chicken to the bag and turn to coat in the buttermilk. Seal the bag and refrigerate 30 minutes or overnight. When you’re ready to cook the chicken, let it sit at room temperature in the marinade for 30 minutes. Pour the oil into a large, straight-sided frying pan or cast iron skillet (the oil should reach halfway up the side of the pan). Heat over medium-low heat until it reaches between 350°F and 360°F on a candy/fat thermometer, about 20 minutes. (If you don’t have a thermometer, test it by sprinkling flour in the oil. If the oil’s hot enough, the flour will sizzle, then quickly dissipate.) Meanwhile, combine the measured flour and the remaining 1 teaspoon salt, 1/2 teaspoon black pepper, pinch of cayenne, and pinch of thyme in a shallow dish. Set a wire rack over a baking sheet and set it aside. When the oil is ready, remove 4 of the drumsticks from the buttermilk, letting the excess drip off. Coat the chicken in the flour mixture, shaking off the excess. Carefully lay the pieces in the oil and fry until cooked through and golden brown, flipping halfway through, about 20 minutes total. (An instant-read thermometer inserted into the thickest part of the drumstick should read 160°F to 165°F. If you don’t have a thermometer, cut a small slit down to the bone; the juices should run clear and there should be no pink at the bone.) Place the drumsticks on the wire rack and season with salt. Repeat with the remaining chicken.