Cut sausage into 1/2-inch slices; set aside.
Combine oil and flour in a 4-quart heavy saucepan; mix until smooth. Cook and stir over medium-high heat for 3 minutes. Reduce heat to medium; cook and stir until roux is a dark reddish brown, 6 to 8 minutes.
Stir in onion, bell peppers, celery, okra and garlic; cook and stir until vegetables are tender, 8 to 10 minutes. Stir in Cajun seasoning.
Pour in broth; bring to a boil. Cook and stir for 1 minute. Reduce heat to low; add sausage, pepper sauce, black pepper and salt. Cover and simmer for 15 minutes.
Server over rice or on its own!