Preheat the oven to 300 degrees F. Pat the brisket dry with paper towels and prick the roast all over with a fork. Cut the brisket in half crosswise into two roasts (just like you'd cut a sandwich or apple in half). Season each roast with salt and pepper.
In a large nonstick skillet, heat the oil over medium or medium-high heat until rippling and hot. Brown the brisket on both sides (working with one roast at a time), about 3-4 minutes per side. Transfer fat-side up to a 9X13-inch baking dish. Repeat with the remaining roast.
Pour off all but 1 tablespoon fat from the skillet. Add the mushrooms and 1/8 teaspoon salt and cook over medium heat until the liquid evaporates and the mushrooms are golden brown, about 6-8 minutes. Add the onions and sugar. Cook, stirring occasionally, until browned, about 8-10 minutes. Add the flour, garlic, and thyme and cook, stirring constantly, until golden and fragrant, about 2 minutes. Slowly whisk in the broths and bay leaves, scraping up any browned bits. Bring the mixture to a boil. Reduce the heat and simmer until thickened, about 6-7 minutes.
Pour the sauce over the roasts in the 9X13-inch pan and cover the dish tightly with aluminum foil. Bake the roasts in the preheated oven until very tender, 4 1/2 to 5 hours. Remove the dish from the oven and let it cool, covered, for 30 minutes to 1 hour, flipping the roasts and recovering the dish halfway through cooling.
Transfer the roasts to a cutting board and trim any extra fat off, if the roasts are excessively fatty. Strain the sauce from the pan through a fine mesh strainer into a fat separator; reserving the mushrooms. Let the liquid settle and then pour into a microwave-safe bowl, leaving the fat behind. Stir in the vinegar and microwave the sauce for about a minute. If the sauce is too thick, thin with chicken or beef broth. Slice or shred the brisket roasts and place on a serving platter with the reserved mushrooms. Pour the warmed sauce over the meat. Serve immediately.
Notes
Brisket: if you are having trouble finding a piece of brisket smaller than half a cow, simply ask the meat counter clerk or butcher to help you. Generally, they can provide you with a smaller cut of brisket. Flat-cut brisket is easier to find and a bit more uniform in texture than point cut (according to Cook's Country). Doubling: this recipe is easily doubled with fantastic results.