Coat a large roasting pan with olive oil and bring to a high heat. Season the brisket generously with salt. Add the brisket to the pan and brown well on both sides. Remove the brisket from the pan and reserve.
Preheat the oven to 375 degrees F.
Lower the heat to medium, ditch the excess oil, and add the bacon and reserved skin to the pan with a little more oil. Cook the bacon, stirring occasionally, until the bacon is crispy and has let out a lot of fat. Add the onions and celery. Season with salt and cook until they are very soft and aromatic, 7 to 8 minutes. Add the garlic and cook for another 2 to 3 minutes. Toss in the mushrooms and cook until they are soft and wilted. Add the balsamic vinegar and reduce it by half. Taste it - it will be very vinegary, but that's ok, don't worry. Season with salt, if needed. Lay the brisket on top of the veggie mix and add the chicken stock to the pan until it just covers the brisket. Add the bay leaves and thyme.
Cover the pan with foil and roast in the oven. Roast for 1 hour, check the brisket, and turn it over. Add more stock if the liquid level has reduced. Cover the pan again and return it to the oven for another hour. Remove the foil and roast for 30 minutes more.
Remove the roasting pan from the oven, transfer the brisket to a cutting board, cover with foil, and let rest for 20 minutes.
Skim off any excess fat from the cooking liquid. If the vegetables are swimming in liquid, reduce the liquid on the stovetop over medium-high heat. You want the sauce to be soupy. Taste the liquid and re-season if needed. Slice the brisket against the grain on the bias. Serve topped with the onion-mushroom mixture and sprinkle with chopped chives.