Position your oven rack in the lower third and preheat to 285°F. Combine the flour with a generous pinch of salt and pepper on a large plate or sheet of waxed paper. Thoroughly pat the chuck roast dry with paper towels on all surfaces, then dredge it completely in the seasoned flour mixture, coating every side well.
Place your Dutch oven over medium-high heat and add the butter and olive oil. Once the fats are shimmering and hot (but not smoking), carefully place the roast in the pot. Sear without moving for 5-6 minutes until a deep golden-brown crust forms on the bottom. Flip and sear the opposite side for another 5-6 minutes. Transfer the beautifully browned roast to a plate and set aside.
Lower the heat to medium. Pour 1 cup of beef broth into the hot Dutch oven and use a wooden spoon to vigorously scrape all those gorgeous browned bits from the bottom and sides of the pot—this is where incredible flavor lives.
Add the sliced onions to the pot and sauté for about 3-4 minutes, stirring occasionally, until they begin to soften and turn translucent. Stir in the minced garlic and cook for 1 minute until wonderfully fragrant. Add the tomato paste and stir it into the onions for about 30 seconds.
Add the carrots and celery to the pot, stirring to combine with the aromatics. Cook for another 2-3 minutes, allowing the vegetables to begin softening.
Pour in the red wine (or additional broth) and quickly deglaze the pot again, scraping up any remaining browned bits. Add the remaining 2 cups of beef broth, Worcestershire sauce, fresh thyme sprigs, rosemary sprigs, and bay leaves. Stir everything together and bring the liquid to a gentle simmer.
Nestle the seared roast back into the pot on top of the vegetables, along with any accumulated juices from the plate. The liquid should come about halfway up the sides of the meat—if it doesn't, add a bit more broth. Season the top of the roast with another light sprinkle of salt and pepper.
Cover the Dutch oven tightly with its lid and carefully transfer it to the preheated oven. Braise undisturbed for 2 hours, allowing the roast to become tender in that low, gentle heat.
After 2 hours, carefully remove the Dutch oven from the oven (use oven mitts—it's hot!). Add the potatoes, gently pushing them down into the braising liquid around the roast. Check the seasoning and adjust with more salt and pepper if needed. Cover again and return to the oven.
Continue cooking for another 60-75 minutes, until the potatoes are fork-tender and the meat is so tender it practically falls apart when you touch it with a fork. The internal temperature should reach 195-205°F for that perfect pull-apart texture.
Remove the Dutch oven from the oven and let it rest for 5-10 minutes. Discard the herb sprigs and bay leaves. If desired, transfer everything to a large serving platter, or simply serve straight from the Dutch oven. Garnish with freshly chopped parsley if you like, and ladle that rich, flavorful braising liquid over each serving.