Place chicken, onion, carrot, bacon rinds and clarified butter in stockpot. Cover and cook on medium heat for 5 minutes to steam. Add cold water to cover by 1-inch. Add bay leaf, increase heat and bring to boil, uncovered. Skim foam from surface, reduce heat to simmer, cover and cook for 1 hour.
While chicken is still warm, strip meat off bones and place on plate. Cover chicken meat with a little cooking liquid to keep moist; cool in refrigerator.
Place carcass back in cooking liquid and crush bones. Bring to boil and reduce liquid by half. Cool and skim off fat. Strain 2 1/2 cups for use.
Add milk to strained stock and boil. Knead together flour and butter and add to milk and stock, whisking until smooth. Do not boil. Season with salt and pepper.
Strain liquid from cooled chicken meat into medium saucepan. Add 2 1/2 cups water and rice, cooking until done. Add 1/2-cup asparagus spears and 1/2 shrimp to rice.
Add chicken meat, remaining asparagus and remaining shrimp to sauce. Add wine and heat through gently. To serve, lay chicken and sauce over rice. Dust with parsley and cayenne.