Heat oil in a heavy stockpot or other wide, deep, heavy pot over medium heat until it is shimmering, almost smoking.
Lightly salt the lamb pieces, add them to the pot and sear them, stirring, until golden brown all over, 5-7 minutes. Transfer to a plate and set aside.
Add the sausage links to the pan and sear them, turning as they cook, until browned all over, 5-7 minutes. Transfer the braciole to the plate with the other meats.
Add the onions to the pot and cook, stirring, until softened but not browned, about 5 minutes. Stir in the garlic, basil and oregano, and cook, stirring, for 1 minute. Pour in the tomatoes and water, and add the sugar. Stir and cook until the sauce comes to a boil. Lower the heat to bring the sauce to a simmer. Return the meats to the pot, along with the meatballs.
Cook the sauce, adjusting the heat to keep it at a low simmer, and stirring occasionally to ensure that nothing scorches, until cooked to your desired thickness- it can range from thin and mildly flavored to thick and richly flavored. When done to your taste, 2-3 hours, use tongs to transfer the meats to a platter. Stir the sauce and reduce further, if necessary. Taste, and if necessary, add more salt.
Slice the sausage crosswise into individual portions. Remove the string or toothpicks from the braciole and slice it crosswise into portions. Arrange the meats and meatballs on a platter, including the neck bone pieces for those who wish to pick the meat from them.
Serve the sauce with cooked pasta and grated cheese, passing the platter of meatballs and sliced meats.