Once you have the ground beef crumbled, add onion, celery, and green pepper. Cook stirring occasionally until beef is no longer pink and vegetables are softened. Add chicken livers and garlic and continue to cook 5 minutes.
Add salt, Cajun seasoning, and thyme.
Add reserved chicken broth and scrape the bottom of the pan to release all the brown pieces. Let simmer 1 minute.
Remove bay leaf from rice and add rice to Dutch oven. Stir it into the ground beef mixture. Keep the heat on while you stir. Once it is combined well, remove from heat.