Heat vegetable oil in a large Dutch oven or stockpot over medium-high heat.
Add chicken pieces and sliced andouille sausage; sear until browned on all sides, then remove from the pot and set aside.
In the same pot, add the onion, bell pepper, and celery; sauté for 6 minutes until the vegetables are softened.
Add the minced garlic and Cajun seasoning, stirring constantly for 60 seconds until fragrant.
Pour in the crushed tomatoes and chicken broth, scraping the bottom of the pot to release any browned bits.
Stir in the uncooked rice, thyme, and bay leaves; bring the mixture to a rolling boil.
Reduce heat to low, cover the pot, and simmer for 18 minutes or until the rice is tender.
Return the cooked chicken and sausage to the pot and add the raw shrimp.
Cook for an additional 4 minutes until the shrimp are pink and opaque.
Discard the bay leaves, adjust seasoning with salt and pepper, and garnish with sliced green onions before serving.