Season the beef with salt and pepper. Heat the fat of choice in a Dutch oven or heavy-bottom pot over medium heat. Sear the beef in batches until browned on all sides. Approximately 3-4 minutes per side. Remove to a platter and set aside.
In the same pot, cook the diced salt pork or bacon until crispy—approximately 3 minutes. Remove and set aside, leaving the rendered fat in the pot.
Add the onion, carrots, and celery to the pot. Sauté until softened, about 5 minutes.
If using, sprinkle the flour over the vegetables and stir to coat. Deglaze the pot with the wine, scraping up browned bits from the bottom. Add the bay leaves and thyme. If needed, add up to 1 cup of water to ensure the liquid nearly covers the meat. Return the beef and salt pork to the pot. Nestle the split garlic bulb into the liquid.
Cover the pot with a lid and simmer gently on low heat (or in a 325°F [163°C] oven) for 3 hours, stirring occasionally. The meat should become tender and the sauce rich and flavorful.
After 3 hours, add the pearl onions and mushrooms to the pot for a final 30-minute simmer (or until the onions and mushrooms are tender).
Once the pearl onions and mushrooms are tender, turn off the heat and remove all the solids from the simmering liquid to a platter.
Be sure to remove the bay leaves and discard them.
Once all the solids are on the platter, remove the meat from the marrow bones. (These are the bones that are in the beef shanks.) Also, remove the sinew and set it aside. See "Recipe Notes".
Remove all the bone marrow from the marrow bones and add it back into the liquid along with the sinew. Using a stick blender, purée the entire mixture until smooth.
Once the bone marrow and sinew have been completely puréed into the braising liquid, return all the solids to the pot. As you do this, you can break up some of the larger pieces of meat into bite-sized pieces.
Turn the heat to medium and simmer until all the solids are warmed through.
Taste and adjust the seasoning with salt and pepper. Garnish with fresh parsley.