Preheat the oven to 325°F. Pat the beef neck bones dry and season generously with kosher salt and black pepper. Heat the oil in a large Dutch oven over medium-high heat and sear the beef in batches until deeply browned. Transfer to a plate.
Add the onion, carrots, and celery to the same pot. Cook for 4–5 minutes until the vegetables begin to soften. Add the garlic and cook for another 30 seconds.
Stir in the tomato paste, smoked paprika, and dried thyme. Cook for 1–2 minutes, stirring constantly, until the tomato paste becomes slightly darker and fragrant.
Pour in the red wine and scrape the browned bits from the bottom of the pot. Let it simmer for 2–3 minutes.
Add the beef broth and bay leaf. Return the browned neck bones to the pot. Bring the liquid to a gentle simmer.
Cover the Dutch oven and transfer it to the oven. Braise for 2½–3½ hours, or until the meat is very tender and pulls easily from the bones.
Remove the neck bones and bay leaf. If the gravy is too thin, simmer it uncovered on the stovetop for several minutes until slightly thickened. Taste and adjust the seasoning.
Return the beef to the gravy or spoon the gravy over the meat. Serve hot with mashed potatoes, rice, grits, noodles, or cornbread.