Have the butcher split the ducks lengthwise, remove and reserve the backbones and wingtips, and reserve the giblets. Chop the reserved backbones and wing tips into large pieces.
Arrange the ducks skin side up in a large stainless steel or enameled baking pan and prick them all over with a skewer. Add the backbones and wing tips, the reserved giblets excluding the liver, the quartered onions, carrot, celery and garlic, and brown the mixture in a preheated very hot oven (550 degrees F)for 40 minutes, pouring the fat into a bowl periodically and reserve it.
Transfer the ducks to a plate and pour any remaining fat into the bowl. Add 1 cup of the wine to the pan and reduce the liquid over moderately high heat by half. Add the stock, the cheesecloth bag, and the tomato paste and bring the liquid to a boil. Add the ducks, spoon some of the liquid over them, and braise them, covered tightly with foil, in a moderate oven (350 degrees F), basting them occasionally, for 40 minutes, or until they are tender.
In a skillet, heat 3 tablespoons of the reserved duck fat over moderately high heat until it is hot. Add the potatoes with a pinch of sugar and salt and pepper, to taste. Saute them, turning, for 15 minutes, or until they are just tender, and drain them. In another skillet heat 2 more tablespoons of the reserved duck fat over moderately high heat until it is hot. Add the white onions with a pinch of sugar and salt and pepper to taste. Saute them, stirring, for 12 to 15 minutes, or until they are browned lightly and just tender, and drain them.
Transfer the ducks to a plate and strain the cooking liquid into a stainless steel or enameled casserole. Return the ducks to the pan and roast them, uncovered, in a preheated hot oven (400 degrees) for 10 to 15 minutes, or until the skin is crisp. Skim the fat from the cooking liquid and reduce the liquid over moderately high heat to about 2 cups.
In a small bowl, combine the remaining wine and the arrowroot, stir the mixture into the reduced cooking liquid, add salt and pepper to taste. Add the potatoes and onions and simmer the mixture for 5 minutes. Quarter the ducks, arrange them on a heated platter and spoon the vegetables over them.
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