With the rack in the middle position, preheat the oven to 180°C (350°F).
In a Dutch oven or ovenproof pot over medium-high heat, brown half of the beef at a time in the oil. Season with salt and pepper. Set aside on a plate.
In the same pot over medium heat, brown the onion, carrots, celery and garlic for 2 minutes. Add the tomato paste and cook for 1 minute while stirring. Deglaze with the wine and let reduce for 1 minute. Return the meat to the pot. Add the beef stock and thyme. Bring to a boil. Cover. Transfer to the oven and bake for 2 hours or until the meat is tender. Remove from the oven.
Remove and compost any excess fat from the meat as needed. Remove and compost the thyme sprigs. Shred the meat by pressing on it with a wooden spoon. Set aside.
In a large pot of salted boiling water, cook the pasta until very al dente (see tip). Set aside 1 cup (250 ml) of the cooking water. Drain the pasta and add to the pot of meat. Add ½ cup (125 ml) of the reserved pasta cooking water and half (1/2 cup/35 g) of the Parmesan. Reheat everything for 1 minute, stirring to coat the pasta in the sauce. Add more pasta cooking water to thin out the sauce as needed. Adjust the seasoning.
Divide the pasta among four plates. Top with the remaining Parmesan. Sprinkle with the parsley.