Move oven rack to the bottom position. Preheat oven to 325℉. Add the vegetable oil to a large dutch oven. Heat over medium heat until shimmering. Brown the onion and carrot halves and in the hot oil, then remove to the tray.
2 tbsp vegetable oil, 2 large yellow onions, 4 whole carrots
Generously salt and pepper the beef short ribs, then dredge them in flour.
4 pounds bone-in beef short ribs, 1 tbsp salt, 2 tsp black pepper, 1 cup all purpose flour
Brown the ribs on all sides. When the ribs are well browned, remove to a tray.
Stir-fry the mushrooms for about one minute, then remove to the tray.
8 ounces Medium cremini mushrooms
Stir-fry the chopped onions, carrots, garlic and celery until browned.
1 large yellow onion, 1 carrot, 1 celery stalk, 4 garlic cloves
Add the dry red wine and deglaze the pot, scraping up any brown bits from the bottom (the fond). Boil until reduced by half.
2 cups dry red wine
Add the tomato sauce, and stir to incorporate.
8 ounces tomato sauce
Add beef stock to a measuring cup then heat in the microwave on high for 2 ½ minutes. Add the hot stock and Worcestershire sauce then boil until reduced by half again.
3 cups beef stock, 1 tbsp Worcestershire sauce
Use a slotted spoon and a colander to remove the chopped vegetables and discard. Return any collected liquid to the pan.
Return the browned ribs to the braising liquid.
Add the browned mushrooms, carrots and onion halves back into the pot. Add the bay leaves.
3 bay leaves
Cover and braise for 1 hour.
Remove the carrots and set aside, then return the covered pot to the oven. Cook for 1 ½ hours more.
Remove the lid and cook uncovered for an additional 30 minutes before removing from the oven.
Remove the meat, mushrooms and onions, then add to the tray with the carrots.
Serve over mashed potatoes, white rice, or egg noodles.