Pat the short ribs dry with paper towels and generously season all sides with salt and pepper.

Heat the oil in a Dutch oven over medium-high heat. Brown the ribs in batches for about 3–4 minutes per side until well caramelized. Transfer browned ribs to a plate.

Lower heat to medium. Add onion, carrots, and celery to the same pot, cooking until softened and lightly browned, about 6–8 minutes. Stir in garlic and tomato paste, cooking for another 1–2 minutes until the tomato paste darkens slightly.

Pour in red wine to deglaze the pan, scraping up brown bits from the bottom. Let it reduce by half. If using beef stock instead, add a splash and scrape the pan.

Return the ribs to the pot and pour in enough beef stock to come about 3/4 of the way up the ribs. Tuck in rosemary, thyme, and bay leaves. Bring to a gentle simmer.

Cover with a heavy lid and transfer to a 325°F (160°C) oven for 2.5–3 hours, or until meat is fall-apart tender. Alternatively, simmer gently on low heat on the stovetop or cook on low in a slow cooker for 6–8 hours.

Remove ribs and keep warm. Strain braising liquid into a saucepan, pressing on solids to extract flavor. Skim off excess fat if desired. Bring to simmer, then stir in heavy cream and cook gently for about 6–8 minutes until slightly thickened. Remove from heat and whisk in sour cream, lemon zest, chopped parsley, and butter if using.

Nestle the ribs back into the sauce or spoon the sauce over plated ribs. Serve over creamy mashed potatoes, polenta, or buttered egg noodles. Garnish with extra parsley.