Trim off excess fat on the ribs. If you skip this step, you will likely need to skim off exess fat (oil) from the top of the sauce. Pat the rib pieces dry and season liberally with salt and pepper both sides. In the meantime, heat the instant pot to high on 'sautee' setting. You can also do this in pan on the stove. Add oil, and brown the rib pieces on both sides. Remove and set aside.
In the same pan, add a little oil if needed, and add the chopped carrot, celery, and onion. Sautee until they start to soften, loosening any fond that is stuck on the bottom of the pan. Add garlic, and sautee a few minutes longer.
Add tomato paste and mix through. Then add wine and stock. Return the browned rib pieces to the sauce. The sauce should cover at least 3/4 of the way up the ribs. If not, add more liquid (any liquid - wine, tomato, stock, or even water). Add bay leaf.
If using instant pot, pressure cook for 40 minutes an natural release for 10 minutes. If using conventional stove top, bring the sauce to a boil then simmer for 2-3 hours or until the meat is tender.
Remove the ribs and shred the meat; set aside.
Cook your pasta following the package instructions but undercook it (just before it gets to al dente). You will finish cooking the pasta in the sauce.
Remove the bay leaf, and check the consistency. It will be watery. If you want a smoother sauce (fewer or smaller vegetable chunks), blend with a hand blender until you get to your preferred consistency. Add a slurry of corn starch and water and bring the sauce to a boil to thicken it. Taste and season with salt and pepper.
Add the partially cooked pasta and let it finish cooking in the sauce - a minute or so more. The pasta will release some starch, which will also help thicken the sauce. Add chopped parsley and toss everything through the sauce.
To serve, top with shaved parmesan and more parsley. Enjoy!