Adjust oven rack to middle position and heat oven to 350ºF (177ºC).

Heat oil in Dutch oven or brasier over medium-high heat until simmering. Season beef short ribs with 2½ teaspoons salt. Brown beef on all sides, about 8 minutes. (Brown in batches if needed so you don't overcrowd pan). Remove and transfer to a plate. Discard all but 1 tablespoon drippings in pan.

Reduce heat to medium-low. Add onion and carrot to pot with drippings; season with 1 teaspoon salt. Cook, stirring to scrape up fond, until softened, about 3 minutes.

Add tomato paste, garlic, anchovies, oregano, pepper flakes and ½ teaspoon black pepper and continue cooking, stirring frequently, until tomato paste starts to stick to bottom of pot, about 5 minutes.

Stir in ¼ cup broth, scraping up any browned bits, then stir in remaining broth and tomatoes and juices; bring to a simmer.

Add browned beef to pot, cover, and transfer to oven; cook 2 hours.

Uncover and continue to cook until sauce has reduced and beef easily shreds apart, 45–60 minutes.

Remove pot from oven; using tongs, transfer beef to a sheetpan or cutting board and let cool for 5–10 minutes. Using a spoon, skim grease from surface of sauce. Using 2 forks, shred beef into bite-size pieces, discarding bones and any large pieces of fat or connective tissue. Return beef to sauce and season with salt and pepper to taste. (Sauce can be refrigerated for up to 3 days or frozen for up to 2 months.)

Toss with pasta: cook desired amount of pasta to al dente according to package directions. For every 8 ounces of pasta, add 2 cups of sauce to a skillet and heat over low. Transfer al dente pasta to skillet with sauce and toss to evenly coat; add a couple tablespoons of pasta cooking water to loosen the sauce and make it silky (continue doing this until desired sauciness is reached). Cook for a minute or so then enjoy immediately.