Preheat oven to 375 degrees F.
Starting from backbone remove all bones except for maybe the tips of the wings. Reserve bones.
To a medium saucepan over medium-high heat, add about 2 to 3 tablespoons olive oil. Once heated, add the vegetables and saute until onions are translucent. Next add the chicken bones and cover with water by about 1 to 2 inches. Bring the mixture to a boil. Once at a boil, reduce the heat and continue to simmer for about 3 to 4 hours, skimming the top as necessary. Next, remove the saucepan from the heat and strain. Either allow the stock to cool down and refrigerate or keep warm until needed.
Stuff boudin mixture into the chicken and sew back together to reform chicken shape. Place chopped onion in bottom drip pan of a 2 piece roasting pan. Place the chicken, breast-side up, in the roasting pan. Place a small piece of butter under the skin on each chicken breast. Season chicken well with Cajun seasoning and cayenne pepper. Place in oven covered at 375 degrees F for about 45 minutes, then uncover and cook an additional 15 to 30 minutes. Deglaze drip pan with white wine or chicken stock and reduce. Pour into strained chicken stock. In a small bowl combine 1 tablespoon cornstarch with 3 tablespoons cold water and whisk until combined. Whisk the cornstarch slurry into the stock mixture and bring to a boil, allowing the sauce to thicken. If the sauce is too thin, add more cornstarch slurry. If it's too thick, add more water. After chicken has rested, top with the gravy and serve with a side of smothered okra, tomatoes and butterbeans, if desired.