How to Make It
Preheat the oven to 475°. In a small roasting pan, spread half of the chopped onions, 1/4 cup of the parsley, 1 tablespoon of the thyme and the garlic, bay leaves and celery. Prick the duck skin all over with a fork and rub the duck with 2 teaspoons of salt, 1 teaspoon of black pepper and 1 teaspoon of herbes de Provence. Set the duck halves on the vegetables, cut sides down, and roast for 10 minutes. Prick the duck skin again, cover the pan with foil and reduce the oven temperature to 275°. Roast the duck for about 3 hours longer, until the meat is very tender and most of the fat has rendered.
Meanwhile, in a large skillet, cook the backbone, neck and wing tips over low heat until well browned all over. Add the remaining chopped onions and cook over moderate heat until browned, about 4 minutes. Pour off the fat from the skillet and add the tomato paste. Cook, stirring, until it begins to brown, about 3 minutes. Add the white wine and bring to a boil. Add the water, chicken stock and sugar and simmer until reduced to 1 cup, about 1 hour. Strain the stock and skim the fat from the surface.
When the duck is tender, transfer the halves to a work surface. Halve each half; remove any vegetables, pockets of fat and loose bones. Transfer the duck pieces to a rimmed baking sheet, skin side up.
Strain the juices from the roasting pan into a saucepan and skim off the fat; boil the strained juices until reduced to 1/4 cup. Add the strained stock and the olives to the saucepan and simmer for 10 minutes. Season the sauce with salt, pepper and herbes de Provence.
Preheat the broiler. Season the duck with herbes de Provence, salt and pepper. Broil 10 inches from the heat for about 5 minutes, or until the duck is hot and the skin is crisp. Spoon the sauce onto a platter and set the duck on top. Sprinkle with the remaining 2 teaspoons of chopped parsley and 1 teaspoon of thyme and serve.