Rub brisket all over with salt and pepper and allow to sit out at room temperature for 40 minutes. After 40 minutes, the moisture that the salt releases from the meat should be reabsorbed into the meat. If the surface of the brisket is still beaded with moisture, allow the meat to sit out at room temperature a little longer.
Preheat oven to 325° F. Heat the oil in a large Dutch oven over high heat and sear brisket for 5 minutes on each side, or until a nice crust forms. After searing, remove the meat from the pot and set it aside.
Melt butter in the Dutch oven and add onions, garlic, thyme, carrots, and tomato paste and cook over medium heat until onions are soft and slightly browned -- about 5 to 7 minutes.
Return brisket to Dutch oven and add the apricots or prunes, beef stock, and red wine. The brisket should be completely submerged in liquid -- if it isn’t, add water or more stock until it is.
Cover the Dutch oven with a lid and cook in the oven until the meat is falling apart, about 3 1/2 to 4 hours.
When the meat is done, remove it from the Dutch oven, set it on a platter, and cover in foil to keep it warm. Boil the liquid over medium heat until it reduces to approximately 4 cups -- this should take about 25 minutes.
After it has reduced, strain the sauce through a sieve and taste it to adjust seasoning.
Cut brisket against the grain in 1/4-inch strips and serve with sauce and stewed apricots or prunes, onions, and carrots.
Cara Nicoletti is a butcher and writer living in Brooklyn, New York. Cara started working in restaurants when she moved to New York in 2004, and was a baker and pastry chef for several years before following in her grandfather and great-grandfathers' footsteps and becoming a butcher. She is the writer behind the literary recipe blog, Yummy-Books.com, and author of Voracious, which will be published by Little, Brown in 2015. She is currently a whole-animal butcher and sausage-making teacher at The Meat Hook in Williamsburg.