Pat the beef dry with paper towels. In a large bowl, mix the flour, 1 teaspoon salt, black pepper, and paprika. Add the beef and toss until it is lightly coated.
Heat 3 tablespoon olive oil in a large heavy pot over medium-high heat. Sear the beef in batches until browned on all sides, about 3 to 4 minutes per side. Add more olive oil as needed between batches. Transfer the browned beef to a plate.
Reduce the heat to medium. Add the onion and celery to the same pot. Cook for about 6 minutes, stirring often and scraping up the browned bits from the bottom, until they begin to soften and build flavor.
Stir in the carrots and cook for 3 to 4 minutes, until they begin to soften slightly and pick up flavor from the pot.
Add the garlic and tomato paste. Cook for 1 minute, stirring often, until the tomato paste darkens slightly and smells rich.
Pour in the red wine and crape the bottom of the pot well to release all the browned bits. Let it simmer for 2 to 3 minutes.
Add the beef broth, Worcestershire sauce, dried thyme, Italian seasoning, bay leaves, and the browned beef with any juices from the plate. Bring to a gentle boil, then reduce the heat to low. Cover and simmer gently for 1 hour 45 minutes.
Add the potatoes and 1/3 teaspoon salt. Cover again and simmer gently for 1 hour, or until the beef is fork-tender and the potatoes are soft but not falling apart.
Remove the bay leaves. Taste and adjust the seasoning if needed. Stir in the butter for a richer, glossy gravy, then sprinkle with fresh parsley before serving.