Heat a large Dutch oven or heavy-bottomed pot over medium-high heat. Add the butter and vegetable oil.
Add the chicken thighs and andouille sausage, browning them on all sides until they develop a beautiful crust.
Remove the chicken and sausage from the pot and set aside.
In the same pot, sauté the onions, bell peppers, and celery until they become soft and translucent.
Add the minced garlic and Creole seasoning to the pot and cook for another minute.
Add the diced tomatoes, chicken broth, parboiled rice, salt, thyme, black pepper, and bay leaves. Bring the mixture to a boil.
Reduce the heat to low, cover the pot, and let it simmer for about 20 minutes, or until the rice is cooked and has absorbed the flavors of the broth.
Uncover the pot and add the shrimp, chicken, and sausage back into the mixture. Stir everything together gently.
Cook for about 5 minutes, or until the shrimp are cooked through and turn pink.
Remove the pot from the heat and let it rest for a few minutes before serving.