Clean the vegetables. Cut the parsley root, celery root, carrots, and leeks into roughly 1 cm cubes.
Heat a generous amount of olive oil in a large, heavy-bottomed pot or Dutch oven over medium-high heat. Add the cut vegetables and cook for 3-4 minutes, stirring occasionally, until they lightly brown.
Pat the beef dry and season generously on all sides with salt and coarsely ground black pepper.
In the same pot, add more olive oil if needed. Add the seasoned beef and sear over medium-high heat for 2-3 minutes per side until browned.
Add the garlic cloves and fresh thyme sprigs to the pot. Stir for about 30 seconds to release their aromas.
Pour 0.5 liters of the red wine into the pot and bring to a boil over medium-high heat. Add the remaining 0.5 liters of wine so the beef is at least halfway submerged in liquid.
Cover the pot tightly with a lid. Place on the middle rack of the preheated oven and braise for at least 3.5 hours at 180°C (356°F) until the beef is very tender.
Remove the pot from the oven and carefully transfer the beef to a cutting board. Wrap loosely in foil and let rest for at least 15 minutes.
While the beef rests, strain the cooking liquid and vegetables through a sieve into a bowl, pressing gently on the solids to extract all liquid.
Return the strained liquid to the pot. Add the sauce thickener or flour-butter mixture and whisk over medium heat for 2-3 minutes until the sauce reaches your desired consistency.
Cut the rested beef into slices and sprinkle with thyme leaves. Serve with the sauce.