Heat 1 tbsp of the oil in a large saucepan or Dutch oven over medium-high heat. Season the beef with 1/2 tsp salt and 1/4 tsp pepper and cook, turning occasionally, until browned on all sides, 3 to 5 minutes. Transfer to a plate.
Heat the remaining tablespoon of oil in the saucepan over medium-high heat. Add the onion, carrots, garlic, rosemary, and 1/4 tsp each salt and pepper and cook, stirring occasionally, until softened, 6-8 minutes. Add the tomato paste and cook, stirring, until slightly darkened, about 1 minute more.
Return the beef to the saucepan and add the chicken broth and tomatoes. Simmer, covered, stirring occasionally until the beef is fork-tender, 60-75 minutes.
Add the mushrooms to the saucepan and cook, covered, stirring occasionally until tender, 10-12 minutes more. If sauce is too thin, simmer, uncovered, until thickened to the desired consistency.
Meanwhile, cook the pasta according to the package directions; drain and return it to the pot. Add the beef ragu and Parmesan and toss. Serve with additional Parmesan.