In a large skillet or Dutch oven, heat the vegetable oil over medium heat. Add the diced onion, bell pepper, and celery. 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.
Add the chicken pieces to the skillet, followed by the Cajun seasoning, salt, and black pepper. Cook until the chicken is browned on all sides, about 5-7 minutes.
Stir in the diced tomatoes (with their juice), chicken broth, and Worcestershire sauce. Bring the mixture to a simmer.
In a small bowl, mix the flour with a few tablespoons of the sauce from the skillet to create a slurry. Gradually stir the slurry back into the skillet.
Reduce the heat to low and let the étouffée simmer for 20-25 minutes, stirring occasionally, until the chicken is cooked through and the sauce has thickened.
Stir in the chopped parsley and green onions just before serving.
Serve the chicken étouffée over cooked white rice.