Heat a drizzle of oil in a heavy skillet or oven-proof pot set over medium-high heat. Season the ribs with salt and pepper and brown on all sides; set aside. Add the onion, carrot and celery to the pan and cook for a few minutes, until they start to soften and loosen any browned bits from the bottom of the pan. Add the garlic and cook for another minute. Pour in the balsamic vinegar and cook for a minute, scraping up the browned bits from the bottom of the pan.
If you're going to braise the ribs in the pot, return them to the pot with the vegetables. Otherwise transfer the veggies to a pressure cooker or slow cooker. Pour the wine overtop, and add enough stock to come about halfway up the sides of the ribs.
If you're cooking them in the pot, cover and cook at 300F for 3 hours. In the pressure cooker, bring it up to temperature according to the manufacturers' directions and cook for about 45 minutes; cool and release the lid. In a slow cooker, cover and set on low for 8 hours.
Strain the veggies out of the sauce or puree them right in the pot with a hand-held immersion blender; bring to a simmer on the stovetop and cook until slightly reduced, if it needs it.