Heat the cast iron or Dutch oven pan on the stove over high heat. Preheat the oven to 275℉.
Generously salt and pepper the ribs. Peel and chop the carrots, clean and chop the celery and onion, and peel all the cloves of garlic in the head.
When the pan is hot and water dances across the pan, add the avocado oil and let heat for 10 seconds. Add the ribs a couple at a time and sear on high heat on all sides. Don't skip this step. It adds flavor and helps lock in the juices. Continue searing and turning the ribs until complete. Set the ribs aside.
Turn the heat down to medium and add the peeled garlic, celery, onions, and carrots to the pan. Saute 4-5 minutes.
Add red wine and deglaze the bottom of the pan if needed (scrape off the cooked bits). Simmer for about 5-6 minutes or until the wine is reduced by half.
Stir in the beef stock, 3 Tbsp. of tomato paste, and 1 Tbsp. of fish sauce. Heat to a simmer.
Return the ribs to the pan (along with any juices), bone side up so the meat is submerged. Add the bay leaves, parsley, and rosemary.
Place the pan in the oven uncovered 3 - 4 hours until the beef ribs are fork-tender. Check the ribs after 90 minutes - 2 hours to ensure the meat isn't getting too brown. If so, cover with foil and continue cooking.
When the beef ribs are fork-tender, remove the pan from the oven and transfer them to a plate and tent with foil to keep them warm. Pour the remaining braised juices through a fine sieve catching the strained juice in a bowl. Discard the cooked vegetables and herbs.
Pour the juices into a sauce pan and heat over medium-low heat. Mix 3 Tbsp. of the juices with 3 Tbsp. of arrowroot powder to make a slurry. Add to the pan and whisk often until the sauce thickens to a gravy. Taste and season as desired.
Serve the braised beef short ribs drizzled with gravy, mashed potatoes and a vegetable for a delicious meal.