Heat the oil in a wide, heavy 4- to 5-quart pot over medium-high heat. Add the beef and cook for 3 minutes, breaking it into small crumbles. Add the onion and cook for another 5–7 minutes, stirring, until the onion softens and the beef is browned. Spoon away excessive pooled fat if necessary.

Lower the heat to medium. Add the garlic, tomato paste, oregano, salt, and pepper. Stir for 1 minute, taking care not to burn the garlic.

Pour in the tomato sauce and broth. Scrape the bottom with a wooden spoon to release browned bits. Bring to a simmer, then stir in the dry shells.

Cover and simmer gently for 10–15 minutes, stirring every 2–3 minutes and scraping the bottom. Begin tasting the pasta after 10 minutes. If the liquid is nearly absorbed while the shells are still firm, add 1/4 cup hot water at a time. If the shells are nearly tender and the sauce is very loose, uncover for the last few minutes. Cook until the shells are tender with a slight bite. Ground beef must reach 160°F/71°C.

Turn the heat to low and stir in the cream. Warm for 1–2 minutes without a hard boil. Remove from the heat and gradually stir in the Parmesan and basil. Taste and adjust seasoning.

Rest for 3 minutes. If the sauce tightens, add warm water a tablespoon at a time until creamy and spoonable. Divide into six portions and serve.