In a large skillet, heat the vegetable oil and butter over medium heat until melted.
Add the diced onion, bell pepper, and celery to the skillet. Sauté for about 5 minutes, or until the vegetables are softened.
Stir in the minced garlic and cook for an additional 1 minute until fragrant.
Sprinkle the flour over the vegetable mixture and stir continuously for about 2-3 minutes to create a roux, cooking until it turns a light golden brown.
Gradually whisk in the chicken broth, ensuring there are no lumps.
Add the Cajun seasoning, thyme, salt, black pepper, and cayenne pepper. Stir well to combine.
Bring the mixture to a simmer, then reduce the heat to low. Cook for about 10-15 minutes, stirring occasionally, until it thickens slightly.
Stir in the shredded chicken and diced tomatoes. Cook for an additional 5 minutes until heated through.
Serve the chicken etouffee over cooked white rice and garnish with chopped green onions.