Mix the ground coffee, cardamom, black cardamom, and salt in a small bowl and rub into the brisket. Cover loosely with plastic wrap and refrigerate overnight.

Preheat the oven to 475°F. Set a rack inside a roasting pan. Put the brisket on the rack and roast until the exterior has browned, about 20 minutes. Lower the oven temperature to 300°F.

Warm the oil in a large skillet over medium heat and add the onions, carrots, and garlic. Cook, stirring occasionally, until the vegetables have softened but not browned, about 8 minutes. Add the tomato paste and cook until it reduces slightly, about 2 more minutes.

Transfer the vegetables to the roasting pan with the rack removed. Add the brisket, dried apricots, brewed coffee, and eggs in their shells. Add enough water to bring the liquid halfway up the side of the brisket.

Cover the pan tightly with two layers of foil, return to the oven, and braise for 1 hour. Remove the eggs, gently tap them all over with a spoon to make a network of small cracks, and return them to the braise. Cover and continue cooking until the brisket shreds easily with a fork, about 2 ½ to 3 more hours, depending on its size. Let the brisket cool in its braising liquid, then refrigerate overnight.

To serve, preheat the oven to 350°F. Slice the cold brisket, return to the braising liquid, and bake until warmed through, about 30 minutes. Spoon the broth over the meat. Serve with the peeled eggs and grated fresh horseradish.