Rinse oxtails and pat dry; trim excess fat if desired
Season generously with salt, black pepper, garlic powder, onion powder, paprika, and thyme (add cayenne if you want heat)
Heat oil in a Dutch oven or heavy pot over medium-high heat
Brown oxtails in batches, 3–5 minutes per side; remove to a plate
Lower heat to medium; add chopped onion, cook 3–5 minutes
Add minced garlic (about 3–4 cloves) and cook 30–60 seconds
Add tomato paste (optional) and cook 1 minute
Deglaze the pot with broth or water, scraping up browned bits
Return oxtails to the pot
Add enough beef broth/stock to mostly cover (about 2–3 cups, depending on pot size)
Add bay leaves and optional vegetables (carrots, celery, bell pepper)
Bring to a simmer, then cover and cook until tender
Simmer on low on the stovetop 2.5–3.5 hours, stirring occasionally
Or cook in a 300°F / 150°C oven 2.5–3.5 hours, covered
For extra richness, skim excess fat during cooking if needed
Check tenderness: meat should pull easily from the bone
Remove oxtails; taste and adjust seasoning (salt, pepper, acid like a splash of vinegar or lemon juice if desired)
Thicken sauce if needed: simmer uncovered 10–20 minutes, or stir in a slurry (1–2 tbsp cornstarch mixed with 2–3 tbsp cold water)
Return oxtails to the sauce to warm through 5–10 minutes
Serve hot with rice, mashed potatoes, pasta, or crusty bread