Preheat the oven to 300°F. Mince two of the cloves of garlic, and set aside.
In a small bowl place the porcini mushrooms and add the boiling water. Let the mushrooms soak for 20 minutes, then remove them, squeeze out any extra water back into the bowl, then strain the soaking liquid through a fine mesh strainer into a large bowl. Chop the soaked mushrooms, and set the mushrooms and the drained soaking liquid aside.
Meanwhile, heat a large deep pan or stockpot over medium high heat. Add the 2 tablespoons olive oil. Season the brisket on both sides with salt and pepper. Sear the brisket on both sides for about 4 minutes per side, until browned. Transfer the brisket to a large plate.
Return the pan with the oil to medium heat. Add the onions, season with salt and pepper, and sauté for 5 minutes until softened and lightly browned. Add the flour and stir until the onions are well coated with the flour, and the flour has started to color, about 1 minute. Add the wine and stir, scraping up any browned bits from the bottom, until the wine is reduced by half. Add the beef broth, the chopped soaked mushrooms, the strained cooking liquid, and the thyme to the pot and bring to a about halfway submerged. Tuck the garlic cloves in their skins around the meat into the liquid. Cover the pot and transfer it to the preheated oven.
Cook for 3 to 3 1/2 hours until the brisket is fork tender. Remove the brisket from the pot to a cutting board with a mot and let it rest for at lest 15 minutes. Let the braising liquid rest in the pot.
Meanwhile, heat the 2 tablespoons in a very large skillet over medium-high heat until melted. Add the mushrooms and garlic, season with salt and pepper, and sauté until the mushrooms have turned nicely brown, and any liquid that was released has been evaporated, about 8 minutes. Sprinkle over the balsamic vinegar and stir to release any browned bits from the bottom of the pan.
Skim and discard any fat that has accumulated on the braising liquid. Stir the mushrooms into the pot.
When the meat has finished resting cut it into slices as thin or thick as you like, across the grain. Transfer the slices neatly back to the pot, and nestle them beneath the sauce and mushrooms. Serve hot. Alternately you may place the meat on a serving platter with sides and ladle the sauce with the garlic cloves and mushrooms over the top.