Step 1Cut the chicken into 10 pieces (2 drumsticks, 2 thighs, 2 wings and 4 breast pieces). Trim any gobs of fat, especially from the edges of the breast, they tend to burn. Save the back and fat for stock.
Step 2In a shallow bowl, toss the chicken parts with the pepper (allow about 1 1/2 teaspoons per pound of chicken), thyme leaves (about 1 teaspoon thyme leaves per pound), and sea salt and toss well. Cover loosely and refrigerate.
Step 3After about 2 1/2 hours, rinse off the salt in cold running water. Try to keep as much of the pepper and thyme as you can (the pepper and thyme will tend to cling). Drain the chicken well and place in a baking dish just large enough to hold all of the chicken in a single crowded layer. Add cold milk to barely cover. Stir to coat all of the chicken and spread the pieces in a single layer.
Step 4Leave the chicken at room temperature for 2 hours, stirring a couple of times to encourage even de-salting.
Step 5Dredge the chicken in flour, lifting the chicken pieces directly from the milk so they are very wet and will hold a lot of flour. Make sure the skin is neatly stretched over the muscle in a natural position. Tap lightly to shake loose stray flour and place on a cooling rack on a baking sheet so that the pieces are barely touching. Refrigerate overnight, uncovered.
Step 6Before cooking, bring the chicken to room temperature to speed up cooking and encourage even browning.
Step 7In a cast iron pan, heat the peanut oil over medium-high heat. (As you add the chicken, the oil level will rise. If the chicken is ever more than half submerged, ladle out some of the oil.) Test for temperature by dipping the edge of the chicken into the oil -- it should sizzle modestly but immediately.
Step 8Add the chicken pieces, tapping off excess flour before placing them in the oil. Don't crowd or overlap the chicken pieces. If the pan is not large enough, fry the chicken in two batches. Don't worry that the coating is sticky. Start with the thighs, then the drumsticks, then wings and upper breasts, and finish with the breast tips. This is the rough order of how long it will take them to cook.
Step 9Adjust the heat slightly as necessary to maintain a discreet sizzle. If the oil gets too hot, reduce the flame slightly. You can also add a few tablespoons of cool oil to the pan, being sure not to pour it onto the chicken pieces. If a piece is browning unevenly -- say the tip of the drumstick is browning too fast -- or only part of a piece is pale, you can prop the piece against the side of the pan so that the done part sits above the oil, or, so long as all the pieces have set a good crust, you can prop one piece against another so that only the part you want to keep browning is submerged.
Step 10Use tongs to turn the chicken, not a fork, which would pierce the skin. Turn when the cooked side is pale gold, about 9 minutes. Don't assume that all pieces will brown evenly -- the pan may not transfer heat evenly throughout.
Step 11Brown the other side in the same way, then turn back over one or two more times to refine the browning of both sides. The curing helps the chicken retain moisture, so there is little harm in leaving the pieces in the hot oil an extra minute or two to get the tastiest, crispiest golden crust.
Step 12Set the chicken on paper towels to drain. Don't stack it -- you just made a perfect crust, don't let the steam destroy it. Serve immediately.