Preheat the oven to 275F.
Set a heavy, ovenproof pot over medium-high heat and add a drizzle of oil. Pat the beef dry with paper towel and sprinkle with salt and pepper; brown the meat on all sides in the pot. Push it to the side (or remove it and set it aside) and add the onion to the pot; cook for a few minutes, until starting to turn golden. Add the garlic and cook for another minute.
Return the meat to the pan and add the stock, barbecue sauce, wine and Worcestershire. Cover and braise in the oven for 3 hours - or put everything into a slow cooker and cook on low for 7-8 hours.
When the meat is done, pull it apart with two forks right in the pot, and turn the oven up to 425F. Put the potatoes on a heavy baking sheet, drizzle generously with oil, toss with your hands to coat the wedges well and spread out in a single layer. Sprinkle with salt. Bake for 15-20 minutes, or until golden.
Top the fries with warm beef and sauce, cheese curds and green onion. Serve immediately.