Heat oil in a large heavy pot over medium heat.
Add flour and stir continuously to make a roux until it turns deep brown.
Add onion, celery, and bell pepper and cook until softened.
Stir in minced garlic and cook for 30 seconds.
Add sausage slices and cook for 3–4 minutes.
Pour in chicken broth and stir to combine.
Add shredded chicken, Cajun seasoning, thyme, bay leaves, salt, and pepper.
Bring to a boil, then reduce heat and simmer for 35–40 minutes.
Remove bay leaves and adjust seasoning if needed.
Serve hot over cooked rice and garnish with parsley.