In a large pot, heat the vegetable oil over medium heat. Whisk in the flour to make a roux, stirring constantly until it turns a dark brown color, about 15 minutes.
Add the chopped onions, celery, green bell pepper, and garlic to the roux. Cook until vegetables are softened, about 5 minutes.
Stir in the chicken broth slowly while stirring to avoid lumps. Add the Cajun seasoning, dried thyme, and bay leaves.
Add the chicken thighs and sliced sausage to the pot. Bring to a simmer and cook uncovered for 45 minutes, until chicken is tender.
Remove bay leaves. Serve gumbo hot over cooked white rice.
Notes
For extra heat, add hot sauce or cayenne pepper to taste. This gumbo freezes well for up to 3 months.