Cook the Rice: Bring 2 cups of chicken broth to a boil in a medium saucepan. Add rice, cover, reduce heat to low, and simmer for 15-20 minutes or until rice is cooked and liquid is absorbed. Fluff with a fork and set aside.
Cook the Meat: Heat oil or bacon grease in a large skillet over medium heat. Add chopped chicken liver (if using) and cook until browned. Add ground beef and pork; cook until browned and cooked through. Drain excess fat if necessary.
Sauté Vegetables: Add onion, bell pepper, celery, and garlic to the skillet. Cook for about 5 minutes until vegetables soften.
Add Spices: Stir in Cajun seasoning, thyme, paprika, cayenne, salt, and pepper. Mix well to combine.
Combine Rice and Meat Mixture: Add cooked rice to the skillet. Stir to evenly combine. Cook on low for 3-5 minutes to allow flavors to meld.
Garnish and Serve: Sprinkle with sliced green onions and chopped parsley if desired. Serve hot.