Prepare the stuffing: blend liver, sausage, ham, and foie gras in a bowl. Add shallot, parsley, eggs, Armagnac, garlic, salt, and pepper. Soak the bread in milk until softened, then remove and gently squeeze to remove excess milk. Add soaked bread to the meat mixture, and mix well. Cover and refrigerate until needed.
Season inside of capon well with salt and pepper. Fill the cavity with stuffing and sew closed with butcher's twine, making sure twine is very tight. (Or, roll stuffing into a log in a double layer of cheesecloth and tie ends closed. Add to bouillon about 1/2 hour before bird is done cooking, and poach until firm.)
Add turnips, onions, carrots, celery, and leeks to a large, deep pot. Place stuffed capon on vegetables, add bay leaves, thyme, and garlic. Add a generous sprinkling of salt and pepper and pour in the stock. It should completely cover bird and vegetables. Add a little water, if needed. Bring liquid to a boil, reduce heat, and simmer gently until meat is very tender and almost falling off the bone, at least 2 1/2 hours. Cook the cabbage in the bouillon for the last 30 minutes of cooking time. Reserve 1 cup of the bouillon for the Gros Sel Sauce.
While the capon cooks, prepare the Gros Sel Sauce.
Carefully remove capon and vegetables from the pot with a large, slotted spoon. Place on a platter and lightly tent with foil. Strain broth, return to pot, and bring to a boil for 5 minutes. Taste to adjust seasoning. Add pasta, and cook until tender, about 8 to 10 minutes. While pasta cooks, cut capon into serving portions. Remove the stuffing from the cavity and cut into slices (or unwrap and slice). Arrange vegetables around capon and cover with foil. Keep in a warm oven.
Ladle the broth with vermicelli into bowls and serve. Don't forget to pass a bottle of red wine for the chabrot. Following the broth, serve the stuffing, vegetables, and capon, pouring some of the hot broth over them before serving. Garnish with capers, pickled onions, cornichons, and mustard. Serve with Gros Sel Sauce.
Combine eggs, mustard, vinegar, and herbs in a bowl. Slowly stir in oil, then add the broth. Season, to taste, with sea salt.