1. Prepare the ducks for roasting: Cut off the wing tips and set aside with the necks, hearts, and gizzards.

Chop the liver and set aside if you would like to use it in the sauce (see step 6). Remove the fat from the tail section and set aside. Truss the ducks (see “How to Truss a Bird,” below). (If you are making the sauce ahead of roasting the ducks, wrap the ducks in plastic wrap and refrigerate.)

2. Cut the necks, wings, hearts, and gizzards into small pieces. If you do not have a nonstick roasting pan, save the wing tips to use as a makeshift roasting rack (see step 11).

3. In a medium saucepan, heat some of the duck fat over medium heat. Pour out all but about 2 tablespoons of the rendered fat. Add the cut up parts and giblets and cook over medium-high heat until browned, about 5 minutes.

Add the onion and carrot and cook until lightly browned, about 3 minutes. Reduce the heat to medium, add the flour, and brown well, being careful not to burn the onion, about 3 minutes.

4. Add the wine and stock and bring to a boil, stirring. Reduce the heat to medium-low and add the shallots, garlic, orange zest, bouquet garni, tomato paste, glace de viande, salt, and pepper. Simmer until the sauce is thick enough to coat a spoon, 45 minutes to 1 hour, skimming frequently to remove all fat and impurities.

5. Meanwhile, make the sweet-and-sour caramel flavoring: In a small heavy saucepan, moisten the sugar with the water. Bring to a boil over high heat and cook, without stirring, until a deep amber color–dark, but not burned. Quickly pour in the vinegar–it will spatter violently for a second–and allow the mixture to boil for a moment. Remove from the heat.

6. When the orange sauce has finished simmering, remove from the heat and stir in the chopped liver, if using. Strain the sauce through a fine-mesh sieve. Taste and adjust the seasoning, if necessary.

7. Add about half of the caramel flavoring to the orange sauce. Taste and add more, if necessary, to offset the bitterness of the orange zest. Stir in the Madeira and the 1 tablespoon Cognac. (The sauce can be made up to 3 days ahead. Cover with plastic wrap to prevent a skin from forming, and refrigerate. If the sauce was made with the liver, reheat in a water bath [bain-marie]. If not, simply bring to a simmer before serving.)

8. With a vegetable peeler, remove the zest from 2 of the oranges and cut into very fine julienne. In a small saucepan, cover the julienne zest with water and bring to a boil. Drain and reserve the julienne.

9. Peel and section all of the oranges, making sure they are completely free of all membranes. Pour off any juice that collects. (The orange can be prepared ahead of time. Cover and refrigerate.)

10. Preheat the oven to 475°.

11. Prick the ducks all over with a or to allow the fat to drain during cooking. Place the ducks on their side in a large roasting pan, preferably nonstick, or rest the duck on the reserved wing tips (see “If You Don’t Have a Nonstick Roasting Pan,” opposite). Season lightly with salt and pepper. Place the roasting pan in the middle of the oven and roast for 15 minutes.

12. Reduce the heat to 425° and roast for another 15 minutes, then turn the ducks onto their other sides and with a bulb baster remove as much fat from the pan as possible.

13. Continue roasting for another 30 minutes or so (roasting time is about 15 minutes per pound). For the last 5 to 10 minute of cooking, turn the ducks breast side up for even color. The ducks are done when their cavity juices run clear. Let the ducks sit 10 minutes before carving.

14. Carve the ducks into individual serving pieces and place them on an ovenproof serving platter and let them sit until about 5 minutes before serving time.

15. Preheat the oven to 500°.

16. Add the reserved julienned zest to the sauce and reheat in a water bath (bain-marie).

17. When ready to serve, place the duck in the upper third of the oven for 3 to 5 minutes to reheat. Meanwhile, sprinkle the orange sections with a dash of Cognac and warm them in a covered skillet over low heat, 3 to 4 minutes (save any juices that collect to thin the sauce, if necessary).

18. To serve: Surround the duck with the orange sections and pass the sauce separately.