Preheat the oven to 350° F.
Heat the olive oil in a Dutch oven or large pan over medium heat. Season the short ribs with salt and pepper on all sides. Sear the short ribs on all sides to lock in the flavor, about 3-4 minutes on each side. Remove the ribs to a plate and tent with aluminum foil.
Drain all but 2 tablespoons of the melted fat. Add in the onion, celery, and carrots and season them with salt and pepper. Stir and cook for 3-5 minutes until soft and slightly browned.
Next, add the tomato puree and cook for about 3-5 minutes. This is important as uncooked tomato puree could give the gravy a tart flavor.
Add the entire bottle of red wine. bring to a boil and reduce by half. Return the short ribs along with the juices on the plate. Then add in the beef stock, parsley, thyme, and bay leaves.
Cut each head of garlic in half around the belly, so to speak. Place the 4 halves in the pan.
If you used an oven safe pan that you can cover to cook the ribs, like a Dutch oven, cover it and put it in the oven. Otherwise, transfer everything into a baking pan, cover it, and put it in the oven. Cook for 3 hours.
After removing the pan from the oven, transfer the ribs to a plate and tent with aluminum foil.
Pour the liquid, vegetables, and herbs through the strainer into a 4 quart saucepan. Remove the heads of garlic and any garlic cloves that may have separated during cooking and set aside. Discard the veggies and herbs. Use a spoon to skim the fat off the surface.
Place all of the garlic in a mesh strainer. Using the back of a spoon, press the garlic through the strainer, scraping the garlic off the bottom into the saucepan as you go until all the roasted garlic is pressed through.
Add in the Worcestershire sauce.
Place the saucepan over high heat and bring to a boil. Reduce the liquid by 1/3. Use a whisk to thoroughly incorporate the garlic and prevent sticking to the bottom.
In a small bowl, mix the cornstarch and water. Pour this into the gravy and cook for 1 minute. Once you see the gravy thicken, remove it from the heat.
Pour some of the gravy over the short ribs and enjoy!