Heat the oil in a large pot over medium heat. Gradually whisk in the flour to make a roux. Cook, stirring constantly, until the roux is a dark brown color, about 20-30 minutes.
Add the onion, bell pepper, and celery to the roux. Cook until the vegetables are tender, about 10 minutes.
Stir in the garlic, andouille sausage, and chicken. Cook until the chicken is browned, about 10 minutes.
Gradually stir in the chicken broth, Creole seasoning, and thyme. Bring to a boil, then reduce heat and simmer for 45 minutes.
Add the okra and simmer for another 15 minutes.
Stir in the shrimp and cook until they are pink and opaque, about 5 minutes. Season with salt and pepper to taste.
Serve the gumbo over cooked white rice.