If using chuck roast instead of pre-cut stew meat, cut the beef into 1-inch cubes, trimming off excess fat.
Place the flour, salt, and pepper in a large zip-top bag. Add the beef cubes, seal the bag, and shake to coat the beef evenly. (Alternatively, mix the beef, flour, salt, and pepper in a large bowl and stir to coat.)
In a large Dutch oven or heavy pot, heat 1 tablespoon olive oil over medium heat. When hot, add half of the beef in a single layer. Brown for about 2 minutes, then flip and brown the other side. Transfer to a clean plate.
Add another tablespoon of olive oil and brown the remaining beef in the same way. (Save any extra flour in the bag—you will add it later.) Remove browned beef to the same plate.
Add the remaining 1 tablespoon olive oil, followed by the onion and mushrooms. Sauté for 4-5 minutes, until softened.
Stir in the garlic, tomato paste, and any remaining flour from the bag. Cook, stirring, for 30 seconds to 1 minute.
Pour in the red wine and broth, stirring to deglaze the pot and scrape up any browned bits. Return the browned beef (along with any juices from the plate) to the pot. Stir in the carrots, potatoes, Worcestershire sauce, Italian seasoning, thyme sprigs, and bay leaf.
Bring the stew to a simmer. Reduce the heat to maintain a low simmer, cover, and cook for 1 ½ hours, stirring occasionally, until the beef is tender.
Remove the bay leaf and thyme stems (the leaves will have fallen off). Taste and adjust salt and pepper as needed.
Serve warm with crusty bread or your favorite side.