Preheat the oven to 300 degrees . Season the brisket on both sides with salt. Heat a large Dutch oven over medium-high heat. Place the brisket fat side down in the pot and cook, turning once, until browned and the fat renders, about 10 minutes. Transfer the brisket to a plate. Add the onions and garlic to the pot and stir well to blend in all the browned bits, then remove from the heat. Add the browned brisket, fat side up, to the pot, nestling it into the onions; season with salt. Cover and bake for 2 hours.
Add 2 cups water, the prunes and apricots, tucking them among the onions; cover and bake for 1 1/2 hours more. Remove the pot from the oven, baste the brisket with the cooking liquid and add the sweet potatoes. Cover and bake until the brisket and sweet potatoes are tender, about 30 minutes. Let the meat rest in the braising liquid for at least 30 minutes before slicing.