Combine chicken and water in a Dutch oven; bring to a boil. Reduce heat; simmer 45 minutes. Cover; chill 8 hours. Remove chicken; skim and discard fat from broth, reserving broth. Bone and chop chicken.
Place flour in a 15- x 10- x 1-inch jellyroll pan. Bake at 350° for 1 hour or until very brown, stirring every 15 minutes. Set aside.
Coat a Dutch oven with cooking spray; add oil. Place over medium heat until hot. Add onion and next 4 ingredients; saute until tender. Add thyme and next 3 ingredients. Add browned flour; stir until smooth. Add reserved broth, chicken, canned broth, tomato paste, and sausage. Bring to a boil; reduce heat, and simmer, uncovered, 1 hour. Peel and devein shrimp; add to broth mixture. Cover and simmer 10 minutes or until shrimp turn pink. Discard bay leaves. Serve over rice.