In a large pot, heat vegetable oil over medium heat. Whisk in flour, stirring continuously, to make a dark roux, about 10-15 minutes.
Add the sliced sausage to the roux and cook for about 5 minutes.
Stir in the chicken pieces and cook until lightly browned.
Add the onion, bell pepper, and celery. Cook until they become tender, about 5 minutes.
Add minced garlic, chicken broth, and Cajun seasoning. Stir well to combine.
Bring the mixture to a boil, then reduce heat and let it simmer for about 1 hour, stirring occasionally.
Add sliced okra and simmer for an additional 30 minutes.
Serve gumbo over cooked rice.