Add the chicken livers and gizzards to a saucepan with enough water to cover. Bring to a simmer, then cook over medium heat for 10 minutes. Drain the livers and gizzards and chop them.

Meanwhile, in a large skillet over medium heat, add the sausage and cook, stirring to break up the meat, until its well browned. (Hard sausage such as andouille will need to be chopped before cooking.) Transfer the meat to a plate, and drain off the fat.

Add the oil to the drained skillet. Put it back over medium heat and add the chicken livers and gizzards, onion, bell pepper, celery, scallions, garlic, salt, and black pepper. Cook, stirring, until the vegetables have softened, about 15 minutes. Stir in the cooked rice and continue to cook, stirring occasionally, for 5 minutes more. Serve hot.