In a heavy bottomed pot or Dutch oven, melt butter with the oil over medium high heat. Add flour and stir constantly with a wooden spoon to form a roux. Cook this mixture, continuing to stir, until it develops a color just darker than peanut butter. It should smell nutty and toasty.
Add the bell pepper, celery, onion, and garlic. Stir in to the roux and cook until softened. Add the tomatoes, bay leaves, Cajun seasoning and thyme and continue to cook for 2-3 minutes. Add 3 cups of the broth, stirring to incorporate, and simmer for 30-40 minutes. Add additional broth if needed.
While the étouffée is cooking, bring the rice, 4 cups chicken broth, butter, and Cajun seasoning to a simmer. Cover and cook for about 20 minutes or until all the liquid is absorbed and rice is tender. Fluff with a fork.
Add the shrimp to the étoufee and cook for 5-6 minutes until they are cooked through. Season to taste with additional cajun seasoning, salt, and cayenne pepper
Serve over rice and garnish with green onions.
