Season the brisket evenly on both sides with salt and pepper. Cover and refrigerate for 2 hours. Meanwhile, in a medium saucepan, bring the wine to a boil and continue boiling until it reduces to half its original volume, 15 to 20 minutes.
Preheat the oven to 350°F.
Evenly sprinkle the brisket all over with flour, shaking off excess. Heat a heavy Dutch oven over high heat. Add 2 tablespoons of the oil and when it is almost smoking, turn the heat to medium-high, carefully add the brisket, and sear until well browned, about 5 minutes per side. Transfer the brisket to a platter. Pour off the fat from the Dutch oven and add another 2 tablespoons of oil. Add the garlic, shallots, carrots, celery, and leek, and sauté until glossy and lightly browned, about 5 minutes. Add half of the apricots and all the parsley, thyme, bay leaves, and tomato paste, and continue to cook 1 minute more.
Return the browned brisket to the Dutch oven and add the reduced wine and 2 quarts stock or broth. If the brisket is not completely covered with liquid, add enough extra stock, broth, or water to cover. Bring the liquid to a simmer. Cover the pot and carefully place it in the oven. Cook until the brisket is fork tender, 2 to 2½ hours. Meanwhile, blanch and peel the onions as instructed in the tips, leaving them whole. Cut the remaining apricots into 1/4-inch strips.
In a small sauté pan, heat the remaining 2 tablespoons oil over medium-high heat and sauté the pearl onions until lightly golden, 5 to 7 minutes. Add the apricot strips and pour in the remaining 1/4 cup stock or broth, stirring and scraping with a wooden spoon to deglaze the pan deposits. Reduce the heat and simmer gently until tender, about 5 minutes. Cover and keep warm.
When the meat is done, carefully transfer it to a heated platter, cover with aluminum foil, and keep warm. Boil the liquid in the Dutch oven until it thickens and reduces to about 1 quart, 15 to 20 minutes. Pour it through a fine-mesh strainer into a bowl, taste, and adjust the seasonings if necessary with more salt and pepper.
To serve, use a sharp knife to cut the brisket across the grain into 1/4-inch slices. Arrange the slices on heated serving plates or on a heated platter, spoon half the sauce over it, and garnish with the pearl onions and apricots. Sprinkle with minced parsley and pass the remaining sauce on the side. Serves 8.