In a large bowl, whisk together the buttermilk, eggs, Cajun seasoning, and 1 tablespoon of the kosher salt. Add the chicken pieces and turn to coat them completely. Cover the bowl and refrigerate for at least 4 hours, or overnight for best results.
When ready to cook, pour the oil into a large, heavy-bottomed pot or Dutch oven. Heat over medium-high heat until the oil reaches 325°F (163°C) on a deep-fry thermometer.
While the oil heats, prepare the dredge. In a separate large bowl or a shallow dish, whisk together the flour, black pepper, and the remaining 1 tablespoon of kosher salt.
Remove one piece of chicken from the marinade, letting the excess drip back into the bowl. Thoroughly coat the chicken in the flour mixture, pressing it on to adhere. Shake off any excess flour and set the chicken on a wire rack.
Repeat with 3 more pieces of chicken. Carefully place the 4 pieces into the hot oil, making sure not to overcrowd the pot. The oil temperature will drop; adjust the heat to maintain a temperature of about 300°F (150°C).
Fry the chicken for 12 to 15 minutes, turning occasionally, until it is deep golden brown and cooked through.
The internal temperature of the chicken should register 165°F (74°C) on an instant-read thermometer. Transfer the cooked chicken to a clean wire rack to drain.
Bring the oil back to 325°F (163°C) and repeat the dredging and frying process with the remaining chicken pieces. Let the chicken rest for a few minutes before serving.