In a large bowl, blend the eggs, buttermilk and Cajun spice. Mix well, and then pour over the chicken pieces, mixing and coating well. After coating the chicken, cover and refrigerate overnight, or for a minimum of 4 hours.
Preheat the oil to 300 degrees F in a deep-fryer. Remove the chicken from the marinade and allow to sit on a baking screen for 5 minutes. This will allow any excess marinade to drip off.
Next, in a large bowl, blend the flour, salt and pepper, mixing well. Dredge the marinated chicken in the flour. Coat the chicken well with flour, and then tap off any excess. Add to the heated oil. Fry the chicken in batches until it reaches 165 degrees F on an instant-read thermometer, 13 to 15 minutes. Then remove from the oil and allow any excess to drip off before serving. This will allow the meat to rest before serving. Repeat the cooking process with the remaining chicken. (Hold the cooked chicken in the oven on low temperature if doing in batches.)