Pat the short ribs dry with paper towels and drizzle with about 2 teaspoon of olive oil. Season generously with kosher salt and black pepper on all sides.
In a large Dutch oven or heavy-bottom pot over medium-high heat, pour in 1 tablespoon olive oil and sear the short ribs until golden brown on all sides, about 2-3 minutes per side. Remove the short ribs from the pot.
Turn the heat down to medium and add the chopped celery, onions, and carrots to the pot. Season them with about 1 teaspoon salt and pepper and sauté them, stirring occasionally, until they are turning golden brown, about 5 minutes. Add in the thyme leaves and garlic and cook for two minutes until they are fragrant.
Stir in the tomato paste, then pour in the wine to deglaze the bottom of the pot. Add the beef stock, bay leaves and other dried herbs, canned tomatoes, and the browned short ribs into the pot.
Cover the pot with its lid slightly ajar, turn the heat to low, and simmer for 2-3 hours, stirring occasionally so the bottom of the pot doesn't burn, or until the meat is fork tender.
When the liquid has reduced down and the meat is fork tender, remove the bones, the meat, and the bay leaves. Stir in the heavy cream.
Optionally, turn the heat off and use an immersion blender to smooth out and emulsify the sauce.
Shred the meat with forks in a mixing bowl, then add the shredded meat back in. Stir well, then taste and season as needed.
Serve over pasta, and with freshly grated parmesan.