In a large heavy pot or Dutch oven, heat the vegetable oil and butter over medium heat until melted.

Add the chopped onion, green bell pepper, and celery. Sauté for about 5 minutes until the vegetables are softened.

Stir in the minced garlic and cook for an additional minute until fragrant.

Sprinkle the flour over the vegetable mixture and stir well to combine. Cook for about 5 minutes, stirring constantly, until the mixture turns a light golden brown (this is your roux).

Gradually whisk in the chicken broth, ensuring there are no lumps.

Add the Cajun seasoning, thyme, salt, black pepper, and cayenne pepper. Stir to combine and bring to a simmer.

Reduce the heat to low and let the mixture simmer for about 20 minutes, stirring occasionally.

Stir in the shredded chicken and cook for an additional 5 minutes until heated through.

Serve the étouffée over cooked white rice and garnish with chopped green onions and parsley.