1. In a large pot, heat the oil over medium heat. Add the flour and cook, stirring constantly, until the mixture turns a dark brown color, about 10-15 minutes. This is your roux.
2. Add the onion, bell pepper, and celery to the roux. Cook, stirring often, until the vegetables are tender, about 5 minutes.
3. Stir in the garlic and cook for another minute.
4. Gradually whisk in the chicken broth until smooth. Add the chicken, sausage, Cajun seasoning, thyme, and bay leaves. Bring to a boil.
5. Reduce the heat to low and simmer, uncovered, until the chicken is cooked through and the flavors have melded, about 45 minutes. Stir occasionally.
6. Season with salt and pepper to taste. Remove the bay leaves before serving.
7. Serve the gumbo over cooked white rice.