To Make the Short Rib Ragu
Turn your instant pot/pressure cooker on to Saute mode, adjust to high heat. While it heats up, add in vegetable oil, then pat short ribs dry with a paper towel. Generously season the meat with kosher salt and pepper on all sides.
Add the seasoned short ribs to the hot Instant Pot and sear the meat on all sides until well browned. Remove short ribs from pot and set aside on a plate.
Add pancetta, onion, carrots, and celery to the pot, and saute the veggies until the onions are translucent, softened, and golden, about 5 minutes. Add in garlic and herbs and cook for one to two minutes, stirring to ensure the garlic doesn't burn.
Deglaze the fond bits at the bottom of the pan with the red wine, scraping the browned bits off the bottom with your wooden spoon.
Add the seared short ribs, pureed tomatoes, and beef broth to the pot. Set Instant Pot to the manual setting for 45 mins and seal the lid.
When your ragu is done pressure cooking, let the Instant Pot release the steam for 15 minutes, then carefully turn the vent for manual release until there is no more steam, the pressure valve has fully lowered, and the lid is safe to open.
Remove the bones and bay leaves, then remove the meat and shred using forks. Put the meat back into the sauce. Skim excess fat off the top of the ragu and taste for seasoning, add kosher salt and black pepper as needed.
Make the Pan Fried Gnocchi
Set a large pot of salted water to boil on the stove top over high heat. Once boiling, cook gnocchi according to the package instructions.
While the gnocchi boils, set a nonstick pan over another burner on medium heat with 2 tablespoons of oil.
When gnocchi is done boiling, use a slotted spoon to scoop them out of the water or drain the pot, shake the gnocchi of excess water, and place them in an even layer in the pan. You'll most likely need to do this in batches if you're making all 8 servings of gnocchi at once.
Cook the gnocchi for 2-3 minutes until a golden brown crust forms before flipping over onto their other side to crisp up.
Serve the gnocchi in pasta bowls with a hearty serving of the short rib ragu on top. Sprinkle on more torn basil and serve with grated or shaved parmesan cheese.