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Make the batter In a large bowl, mix the buttermilk with 2 tsp. sea salt and 1/2 tsp. pepper. Add the chicken and toss to coat. Cover and refrigerate for at least 4 and up to 24 hours, turning occasionally.
Make the Broth for the Gravy (optional)
Heat the oil in a large (4-quart) saucepan over medium-high heat. Add the chicken parts and onion and cook, stirring often, until the chicken parts lose their raw color, about 5 minutes. Reduce the heat to low, cover, and cook, stirring occasionally, until the chicken parts release juices, about 20 minutes. Add 1 quart of water and 1/2 tsp. salt. Increase the heat to medium high, bring to boil, and then reduce the heat to maintain a simmer. Simmer until the broth is flavorful, about 20 minutes. Strain the broth into a 1-quart glass measuring cup. (You will need 1-1/2 cups for the gravy; reserve remaining broth for another use.) Set aside.
Fry the chicken When you’re ready to fry the chicken, put the flour, paprika, 2 tsp. sea salt, and 1/2 tsp. pepper in a large doubled brown paper bag, and shake to combine. Working in 2 batches, drop the chicken pieces into the flour mixture, fold the top of the bag closed, and shake to coat completely. Arrange the coated chicken on a large wire rack set over a large rimmed baking sheet. Discard the remaining flour mixture.
Pour enough oil into a deep heavy-duty 12-inch skillet (preferably cast iron) to reach a depth of 1/2 inch. Heat the oil over medium-high heat until a deep fat/candy thermometer clipped to the side of the skillet without touching the bottom registers 350°F.
Carefully arrange the chicken skin side down in the hot oil—it’s fine if the pan is very crowded. The temperature will drop to about 300°F. Partially cover the skillet with a lid or a baking sheet, leaving the thermometer visible, and fry until golden-brown, about 5 minutes, adjusting the heat as needed to maintain 300°F to 325°F. If necessary, move the pieces around for even browning. Turn the chicken over and fry, uncovered, until browned all over and an instant-read thermometer registers 165°F when inserted into the thickest part of each piece, 5 to 7 minutes more.
Meanwhile, wash and dry the wire rack and baking sheet and set the rack over the sheet near the skillet. Using tongs, transfer the chicken to the rack to drain briefly. Serve warm or at room temperature.
Make the Gravy (optional)
Strain the hot oil from frying the chicken into a heat-safe container. Pour 3 Tbs. of the oil back into the same skillet and set it over medium-high heat. Whisk in the flour and cook, whisking constantly, until the flour mixture is golden-brown, 1 to 2 minutes. Whisk in the 1-1/2 cups broth and the thyme and bring to a simmer. Continue to simmer, stirring often, until thickened, 2 to 3 minutes more. Add the lemon juice and season to taste with salt and pepper. Serve immediately with the fried chicken.