Pat the beef dry. Season all sides with kosher salt and pepper.
Heat a large Dutch over medium high heat. Add 2 T olive oil to the pot. Once shimmering add the chuck roast and sear on all sides, 3-4 minutes per side. Do not overcrowd the pan. Work in batches if you need to.
Transfer the seared meat to a plate and set aside. If any drippings are in the pan, drain all but 2 tablespoons. If there are no drippings or olive oil left in the pot, add the remaining 1 T olive oil (or more as needed) and place the pot back on the stove over medium heat.
Soffritto: Add the soffritto (carrots, onion, celery) to the pot. Season with kosher salt and pepper, about 1 tsp each. Stir often. Cook until deeply browned, about 15 minutes. Once browned add the minced garlic and cook for 1 minute more. Add the tomato paste and stir, cooking for another 2 minutes or until the tomato paste is browned
Pour in the 1 cup of beef broth (or red wine) to deglaze the pan. Stir the mixture, scraping up any browned bits from the bottom of the pan. Cook until the broth is almost completely absorbed into the soffritto.
Using kitchen twine, tie the fresh rosemary, sage, and thyme leaves together. Add them to the pot along with the bay leaves, crushed tomatoes, 2 1/2 cups beef broth, and seared beef chuck roast pieces.
Cover the pot and simmer, stirring occasionally for 2 ½ to 3 hours or until the beef is fall-apart tender. If the ragu loses too much liquid, add a splash of water or more beef broth to the pot.
Once the beef is fall-apart tender, transfer them to a cutting board and shred the meat. Remove the tied herbs and bay leaves, discard. Place the shredded beef back into the pot with the sauce.
Boil water in a large pot. Once boiling add a tablespoon of kosher salt and the pasta. Cook until al dente. Once cooked, drain the water, MAKING SURE TO RESERVE 1 CUP OF THE STARCHY WATER.
While the pasta is cooking, bring the ragu back to a simmer. Stir in the heavy cream and parmesan cheese. Add the cooked pasta and toss it in the sauce until fully coated. If needed, add a splash of the starchy pasta water to loosen up the sauce.
Serve immediately with more cheese and fresh herbs, if desired.