In a large saute pan, over medium heat, add the olive oil. When the oil is hot, add the shallots. Season the shallots with salt and pepper. Saute the shallots for 1 minute. Stir in the garlic and tomatoes. Season with salt and pepper. Saute for 2 minutes. Stir in the duck meat. Saute for 1 minute. Stir in the reduction and bring the liquid to a boil. Reduce to a simmer and cook for 3 to 4 minutes. Stir in the butter and parsley. Remove from the heat and re-season if necessary. Serve over mash potatoes or cheesy grits.
In a large saucepan, add the stock, onions, carrots, celery, and bouquet garni. Bring the liquid up to a boil and reduce to a simmer. Simmer until the liquid reduces by half, about 1 1/2 hours. Strain through a fine-mesh strainer and cool. Refrigerate overnight and remove any congealed fat from the surface. The reduction can be stored in the freezer in ice cube trays for 1 month.
Break and crack the carcass. In a large stockpot, heat the vegetable oil. Season the bones with salt and pepper. Add the bones to the pot and brown for about 10 minutes, stirring often. Add the onions, carrots, celery, garlic, and bay leaves. Season the mixture with salt. Cook until the vegetables are soft, about 5 minutes, stirring often. Add the wine and tomato paste and stir to mix. Cook for 5 minutes, stirring occasionally. Add the water. Put the thyme, parsley sprigs, and peppercorns in a piece of cheesecloth, tie it together with kitchen twine, and add it to the mixture. Bring the mixture to a boil. Skim off any cloudy scum that rises to the surface. Reduce the heat to medium and simmer, uncovered, for 3 hours. Strain through a fine-mesh strainer and cool. Refrigerate overnight and remove any congealed fat from the surface. The stock can be stored in the freezer for 1 month.