Place the sliced onion in the bottom of a roasting pan. Remove the giblets from the hens, and rinse them inside and out. Remove any excess fat and leftover pinfeathers and pat the outsides dry. Place the hens on top of the sliced onion.
Pack the cavities of the hens with the cornbread stuffing.
Tie the legs together with kitchen string and tuck the wing tips under the body of the hens. Rub the hens with olive oil and sprinkle with salt and pepper.
Roast the hens for 30 minutes, or until the juices run clear when you cut between a leg and thigh.
Melt the butter in a large saute pan. Add the onion and cook for 5 minutes over medium-low heat, until translucent. Break the cornbread into pieces and place the pieces in a large bowl. Add the onion mixture to the cornbread. Add the celery, chicken stock, parsley, salt and pepper, and mix together.