When ready to cook, set temperature to High and preheat, lid closed for 15 minutes.
Preheat a large cast iron dutch oven in grill with lid closed for 15 minutes.
Pat beef dry. Add vegetable oil to preheated dutch oven and brown beef on all sides, turning with tongs, about 8 minutes. Transfer to a plate and sprinkle with 1/4 tsp sea salt and 1/4 tsp pepper.
Add chopped carrots, onion, and garlic to oil in dutch oven and cook stirring occasionally until softened, about 10 minutes. Stir in 1 cup puréed tomatoes cover dutch oven and close grill. Cook for 20 minutes. Add wine and cook for 30 more minutes with lid off dutch oven (but grill closed), stirring every 10 minutes or until sauce thickens.
Add demi glacé, thyme, bay leaf, vinegars, and remaining 1/2 tsp sea salt, 1/4 tsp pepper and beef along with any juices that have accumulated on plate. Cover dutch oven with lid and turn grill temperature down to 325℉. Braise for 4 hours or until beef is very tender.
While beef is braising, pierce potatoes with a fork a few times and set aside. The last 1-1/2 hrs of cooking, place the potatoes directly on grill great next to dutch oven.
Pull the Dutch oven and potatoes from grill. Let potatoes cool slightly and then cut in half, lengthwise, using a fork to scrape flesh into a large saucepan.
Smash mixture with potato masher until mostly smooth. Stir in butter and half and half, season generously with salt and pepper to taste and set aside.
Pull the beef out of the pot and spoon some of the braising liquid over each rib. Serve with the mashed potatoes and your favorite toppings. Enjoy!
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