Take the short rib out of the refrigerator 30 minutes ahead of time. Season both sides with salt and pepper. Heat the dutch oven to medium-high heat and add olive oil. Sear the short ribs on all sides for 1-2 minutes until they have a golden brown crust. Set them aside. Preheat the oven to 325℉.
Lower the heat to medium. Add the pancetta and saute until golden brown, about 5-7 minutes. Remove and set them aside. Remove some of the oil remaining. Leave about 2 tablespoon of oil.
Add the carrots, onions, and celery. Saute for 5-8 minutes or until they begin to soften. Stir in the garlic and saute for 1-2 more minutes. Then add tomato paste and combine. Saute for another 3 minutes. Add the red wine and simmer for 5 minutes.
Once the wine is reduced by about half, add in the whole plum tomatoes, crushing them by hand or pulsing them in a food processor. Bring the tomatoes to a simmer. Add the basil. Then place the short ribs in the sauce, buried about halfway. Cover the pot, and place it in the oven for 2 hours. Stir after an hour. Then remove the cover and continue cooking for 30-45 more minutes. Once the short rib is fork-tender, remove the pot from the oven.
Remove the short ribs and place them into a bowl. Use two forks to shred the meat. Once completely shredded, return to the pot and mix with the sauce.
Bring a pot of water to a boil. Add your gnocchi and cook as directed.
Add some sauce and the gnocchi to a pan on medium heat. Bring the sauce to a simmer, combine with the gnocchi, and serve with extra sauce on top!
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Notes
Don't move the short ribs while searing. Let them sit there until the crust develops. The more you move them, the more steam you will create, which will toughen the meat.
Don't completely submerge the meat in the sauce. If you don't use a big enough pot, the meat will be completely submerged and will not braise. You want the meat to only be halfway into the sauce.
Check the short ribs after 1 hour of being covered to stir. Then check again after 30 minutes of having the cover off for tenderness.