Dates add just a hint of sweetness to this savory main.
Preheat the oven to 350°. Season the ribs generously with kosher salt and pepper. Heat a cast-iron skillet over high. Add 2 tbsp. oil, then the short ribs. Cook, pressing the ribs with tongs to get a deep brown sear, 2 to 3 minutes per side.
Transfer the ribs to a large Dutch oven or pot. Clean and dry the skillet. Add the remaining 2 tbsp. oil to the skillet and heat over medium-high. Add the carrots, onions, and celery. Cook, stirring occasionally, until golden brown, 7 to 10 minutes. Add the tomato paste. Cook, stirring often, until the vegetables are deep golden-brown, 2 to 3 minutes. Transfer the vegetables to the pot. Add the dates, thyme, and garlic.
In a large saucepan, simmer the stock, wine, vinegar, and mushrooms over medium heat. Stir in 2 tsp. kosher salt. Add the mixture to the pot. Cover and transfer to the oven. Cook until the meat is very tender when pierced with a fork, about 2 1/2 hours. Discard the garlic and thyme sprigs. Spoon off any fat from the surface of the sauce.
Divide the short ribs among plates. Top with the vegetables and some of the sauce. Serve with the remaining sauce and the Fingerling Potatoes with Tarragon & Chive Butter.