Season the fish all over with the salt and pepper. Dredge a piece of fish through the flour and shake off any excess. Test the oil temperature by dipping a corner of the flour-coated fish into the oil — if it sizzles vigorously, the oil is ready; if not, heat the oil for a bit longer. Carefully place the piece of fish in the oil and repeat the flouring process with another 4 to 6 pieces of fish; don't crowd the pan. Fry the fish, turning once, until cooked through and golden brown, about 2 minutes on each side. Transfer the fish to the paper towel-lined plate to drain. Continue frying the rest of the fish in batches.