Heat vegetable oil in a large Dutch oven or stockpot over medium-high heat.
Brown the andouille sausage rounds for 3-4 minutes until fat is rendered, then remove with a slotted spoon.
Add chicken cubes to the pot and sear until browned on all sides, then remove and set aside with the sausage.
In the remaining fat, sauté the onion, bell pepper, and celery for 5-7 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Stir in the Cajun seasoning, dried thyme, and bay leaves.
Deglaze the pot with chicken stock, scraping the bottom to release browned bits.
Add the diced tomatoes with juice, browned sausage, and chicken back into the pot.
Bring the mixture to a boil, then reduce heat to low and simmer for 20 minutes.
Add the uncooked rice to the soup, cover, and simmer for an additional 15-20 minutes until the rice is tender.
Remove bay leaves, adjust seasoning with salt and pepper if needed, and serve hot.