Preheat the oven to 325°F with a rack in the lower third and no racks above to make room for a Dutch oven or similar large, sturdy pot.
Pat the beef dry with a paper towel and season generously on all sides with salt and pepper.
Heat the oil in an oven-safe Dutch oven or similar large, sturdy pot over medium heat. Once shimmering, add the beef, working in batches to avoid crowding the pan. Sear until you have a golden-brown crust on all sides, about 3 minutes per side. Transfer seared beef to a plate and repeat with the remaining beef. Leave any drippings in the pot.
With the pot set over medium heat, add the onions, carrots, and celery and cook, stirring often, until softened, about 5 minutes. Add the garlic and cook for 1 minute, until fragrant. Don't burn it!
Return the beef to the pot along with any juices that have accumulated. Add the wine, scraping up any browned bits on the bottom of the pot. Add the thyme sprigs, broth, and a pinch of salt, and bring to a simmer. Turn off the heat. Cover the dish and carefully transfer to the oven. Braise for 2½ to 3 hours, until the beef is easily pierced with a knife. Remove from the oven.
Fish out and discard the thyme sprigs. Cover the dish and let rest for at least 10 minutes and up to 1 hour before serving. Garnish with fresh thyme leaves. (See notes if you'd like to thicken the braising liquid).
Notes
If you wish to reduce the sauce, remove the braised beef from the pot and cover the meat to keep it warm. Bring the braising liquid to a simmer on the stove and simmer until reduced to your liking. Return the beef to the sauce and warm through before serving.
TO STORE: Refrigerate leftover braised beef in an airtight container for up to 4 days.
TO REHEAT: Rewarm in a Dutch oven on the stovetop over medium-low heat or in the microwave.
TO FREEZE: Freeze leftovers in an airtight, freezer-safe storage container for up to 3 months. Thaw overnight in the refrigerator before reheating.