Heat oven to 275°F. Heat a large dutch oven over medium-high heat and drizzle with the oil. Generously season the short ribs on all sides with salt and pepper. Work in batches to sear the short ribs on all sides for about 6-8 minutes, turning every 2-3 minutes, until a nicely browned crust develops. Transfer the seared short ribs to a plate and set aside while searing the rest of the meat.
2 Tablespoons vegetable oil, 5 pounds bone-in beef short ribs, kosher salt, freshly ground black pepper
Pour off and discard all but 2 Tablespoons of the fat, leaving any browned bits on the bottom of the dutch oven. Reduce the heat to medium and add the garlic, cut sides down. Cook for 1-2 minutes until golden brown. Add the onion, celery, and carrots. Season with salt and pepper, then cook for 5-10 minutes, stirring occasionally, until the vegetables are slightly softened but not browned.
2 large heads of garlic, 1 medium onion, 4 celery ribs, 2 medium carrots
Add the tomato paste and stir well to evenly coat the vegetables. Continue to cook and stir for 2-3 minutes to caramelize the tomato paste.
3 Tablespoons tomato paste
Add the red wine and use a wooden spoon to deglaze the pan by scraping any browned bits off the bottom of the dutch oven. Let it simmer for 3-4 minutes to reduce a bit.
2 cups dry red wine
Return the seared short ribs and any juices to the pan, then add the beef stock and thyme. The meat short be covered by the liquid and you can add more beef stock if necessary. Bring the liquid to a simmer, then cover the dutch oven with a lid and transfer to the preheated oven.
2 cups beef stock, 4 sprigs thyme
Cook for 3½ to 4 hours without uncovering the dutch oven until the short ribs are fall-off-the-bone tender.
Remove the ribs from the pot carefully using tongs (for presentation purposes, you want the bones and the meat intact, but the meat will be so meltingly soft that this is sometimes easier said than done) and transfer to a large serving plate. Sprinkle with the chopped parsley and lemon zest.
1 cup parsley, 1 Tablespoon finely grated lemon zest
Transfer the liquids from the dutch oven to a fat separator and strain off the fat. Serve with the extra cooking liquids on the side for drizzling over the meat, or thicken them into a gravy, if you like.