Preheat oven to 450°. Chop each carcass into 4 pieces; place in a lightly greased jelly-roll pan. Roast, turning occasionally, 35 to 45 minutes or until browned.
Cook onion and next 3 ingredients in hot oil in a Dutch oven over medium-high heat, stirring occasionally, 3 to 4 minutes or until tender. (Do not overcook.)
Add white wine to onion mixture, increase heat to high, and bring to a boil. Boil, stirring occasionally, 3 to 5 minutes or until wine is reduced by three-fourths. Add carcass pieces and water to cover by 2 inches. Return to a boil over high heat; reduce heat to low, and simmer, stirring occasionally, 1 1/2 hours.
Pour sauce through a fine wire-mesh strainer into a small saucepan, discarding solids. Let stand 10 minutes, and skim fat from sauce. Bring sauce to a boil over medium-high heat.
Whisk together cornstarch and 1 Tbsp. water. Whisk cornstarch mixture into sauce. Return to a boil over medium-high heat, stirring constantly. Reduce heat to low, and simmer 10 minutes or to desired consistency. Stir in parsley and chives; add salt and pepper to taste.
*2 to 3 lb. chicken backbones or wings may be substituted.