Combine oil and flour in a large microwave-safe glass bowl.
Microwave on high for 1 minute, then stir vigorously.
Repeat this process, microwaving in 1-minute intervals and stirring between each, until the roux reaches a deep chocolate-brown color (about 8-10 minutes total).
While your roux is developing in the microwave, brown the andouille sausage in a Dutch oven over medium-high heat until edges are crispy (about 5 minutes). Remove and set aside.
In the same pot, add the holy trinity (onions, celery, bell peppers) and sauté until softened, about 4 minutes. Add garlic for the final 30 seconds.
Carefully add your completed dark roux to the vegetables, stirring to coat everything evenly.
Gradually whisk in the warm chicken broth, stirring continuously to prevent lumps.
Add Cajun seasoning, bay leaves, browned sausage, and shredded rotisserie chicken.
Bring to a gentle simmer and cook for about 15 minutes, allowing flavors to meld.
Taste and adjust seasoning with salt and pepper.
Remove bay leaves, stir in green onions and parsley.
