Whisk flour, salt, black pepper, and cayenne pepper together in a bowl; add cubed chicken and dredge to coat. Reserve remaining seasoned flour.

Heat 1/4 cup butter and olive oil in a large skillet; add chicken and cook until golden brown and no longer pink inside, about 10 minutes. Transfer to a bowl, leaving butter mixture in skillet. Reduce heat to low.

Stir 1 tablespoon reserved seasoned flour into pan drippings; cook and stir until mixture forms a paste and dark caramel color, 10 to 15 minutes. Watch carefully and stir constantly to avoid burning. Add celery, bell pepper, onion, garlic, and remaining 2 tablespoons butter to flour paste; toss to coat with butter and browned flour mixture. Cook and stir until vegetables soft, about 10 minutes.

Add water to cover vegetables; add bay leaves and bring to a simmer. Return chicken to skillet; cook until flavors blended, about 10 minutes more. Serve with cooked rice.