Pour off all but 1 tablespoon of oil from the pan. Add the onions and a generous pinch of salt. Cook over moderate heat, stirring occasionally, until browned, 12-15 minutes. Add garlic and cook for an additional minute. Add tomato paste, and oregano and cook, stirring until fragrant, about 2 minutes. Add the crushed tomatoes with their juices and the carrot and bring to a boil.

Return the short ribs and their juices to the casserole, cover partially and simmer over low heat, turning the short ribs occasionally, until the meat is very tender and the sauce is thick, about 4 hours.

Transfer the short ribs to a plate, discard the carrot. Discard the bones and shred the meat into 11/2 inch chunks. Return the meat to the sauce and season with salt and pepper. Keep warm over low heat.

In a large saucepan of salted boiling water, cook the spaghetti until al dente. Drain the pasta, reserving ½ cup of the cooking water. Transfer the spaghetti to the pot with the sauce, toss well and cook for 1-2 minutes. Add a splash of the reserved pasta water if the sauce is too thick. Serve the pasta immediately.