Preheat the oven to 300°F. In a large Dutch oven, heat the olive oil over medium-high heat. Season the short ribs with salt and pepper, then sear them in the pot until browned on all sides. Remove the short ribs and set aside.
In the same pot, add the diced onion, carrots, and celery. Cook for about 5 minutes until softened. Add the minced garlic and cook for an additional minute.
Stir in the crushed tomatoes, beef broth, red wine, tomato paste, thyme, and oregano. Bring to a simmer. Return the short ribs to the pot, cover, and transfer to the oven. Cook for 3 hours, or until the meat is tender and easily pulls away from the bone.
While the ragu is cooking, prepare the mashed potatoes. Boil the cubed potatoes in salted water until fork-tender, about 15-20 minutes. Drain and return to the pot.
Add the butter, milk, and Parmesan cheese to the potatoes. Mash until smooth and creamy. Season with salt and pepper to taste.
Once the ragu is done, remove the short ribs and shred the meat, discarding the bones. Stir the shredded meat back into the sauce.
Cook the pasta according to package instructions. Drain and toss with the ragu. Serve the pasta topped with Parmesan mashed potatoes and garnish with fresh parsley.