Season the steaks liberally with salt and pepper. Preheat a large, cast-iron skillet or griddle over high heat. Drizzle the steaks with EVOO and cook, turning once, for 8-10 minutes for pink centers. Let the meat rest until cool enough to handle, then slice very thinly on a diagonal.
Bring a large pot of water to a boil, salt it, add the pasta and cook until al dente. Drain, reserving a cup of the pasta cooking water.
Meanwhile, in a large skillet, heat 2 tablespoons EVOO, 2 turns of the pan, over medium-high heat. Add the onion, celery, carrot and garlic; season with the rosemary, salt and pepper. Cook until the vegetables are softened, 5-7 minutes. Stir in the tomato paste for 1 minute to wake up the aroma and flavor. Stir in the beef stock, wine and Worcestershire; lower the heat and simmer.
Drop the sliced meat into the gravy, then add the pasta with a splash of the reserved pasta water to combine. Serve in shallow bowls and top with a drizzle of EVOO, the cheese and parsley.