In a large skillet 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 5-7 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 3-4 minutes, stirring constantly, until the mixture turns a light golden brown.
Gradually whisk in the chicken broth, ensuring there are no lumps.
Add the Cajun seasoning, salt, black pepper, and cayenne pepper. Bring the mixture to a simmer and cook for about 10 minutes, stirring occasionally.
Stir in the shredded chicken and diced tomatoes. Simmer for another 5-10 minutes until heated through.
Serve the chicken étouffée over cooked white rice and garnish with sliced green onions.