Reduce heat to medium-low; simmer until chicken is cooked through, 35-40 minutes.
Remove chicken; cool, then shred meat. Strain stock; discard solids and set aside.
Heat butter in an 8-qt. saucepan over medium-high heat. Whisk in flour; cook, stirring, to make a dark roux, 18-20 minutes.
Add remaining chopped onion, celery, and bay, plus bell peppers, garlic, salt, and pepper; cook, stirring, until soft, 8-10 minutes.
Add andouille, Creole seasoning, cayenne, dried thyme, and chile flakes; cook until fat begins to render from sausage, 3-4 minutes.
Stir in 5 1/2 cups of reserved stock (chill remainder for another use); bring to a boil. Reduce heat to medium; cook, stirring, until thickened, 8-10 minutes.
Add chicken; cook until hot, 2 minutes more. Serve with white rice.