1. In a large Dutch oven or heavy-bottomed pot, heat the olive oil over medium-high heat. Season the short ribs with salt and pepper, then brown them on all sides, about 10 minutes. Remove and set aside.

2. In the same pot, add the chopped onion, carrots, and celery. Sauté for 5-7 minutes until softened. Add the minced garlic and cook for another minute until fragrant.

3. Stir in the tomato paste and cook for 2-3 minutes until slightly darkened. Add the crushed tomatoes and beef broth, stirring to combine.

4. Add the thyme, rosemary, and bay leaf. Return the browned short ribs to the pot, ensuring they are mostly covered by the sauce.

5. Cover the pot, reduce heat to low, and braise for 3-4 hours or until the meat is fork-tender.

6. Remove the short ribs, shred the meat with two forks, discarding the bones. Return the shredded meat to the sauce and mix well. Adjust seasoning with salt and pepper.

7. Place the peeled and chopped potatoes in a large pot, cover with water, and bring to a boil. Cook for 15-20 minutes until tender.

8. Drain the potatoes, return to the pot, and add butter and milk. Mash until smooth and creamy.

9. Stir in Parmesan cheese and season with salt and pepper to taste.

10. To serve, spoon mashed potatoes onto plates and top with a generous ladle of ragu. Garnish with fresh parsley if desired.