Preheat the oven to 325°F. Pat the roast dry with paper towels then season generously on both sides with salt and pepper. Heat a heavy saucepan or Dutch oven (with a tight-fitting lid), into which the roast will fit snugly, over medium-high heat. Add olive or grapeseed oil to cover the bottom of the pan. Sear the roast on both sides, adjusting the heat as necessary, to achieve a deep brown crust on the meat. Take your time with this step, you are developing the “fond” that will flavor the cooking liquid. Remove the roast to a plate and pour off all but a teaspoon or so of the fat.
Place the pan over medium heat and add the chopped onion and a large pinch of salt. Saute until the onion is translucent and beginning to brown a little. Toss in the garlic cloves and stir again for a minute or two. Add the vinegar and stir well to dislodge those browned bits on the bottom of the pan. Raise the heat and let the vinegar bubble away until there is only a glaze left in the pan. Add 1 1/2 cups of beef broth and the fresh herbs and bring to a lively simmer. Add the roast back to the saucepan, there should be enough liquid to come up the sides of the meat, but not completely cover it. (Add the remaining 1/2 cup of broth if necessary).
Cover the roast with a circle of parchment paper, put the lid on the pot and place the roast in the oven. Braise the meat for two hours, turning it once, halfway through. Add the carrots and potatoes to the pot and braise for an additional hour or until the vegetables are cooked and the meat is meltingly tender.
To serve immediately remove the meat and vegetables to a warm serving bowl, cover with foil, and set in a low oven. Strain the cooking liquid through a fine sieve, pushing hard on the solids. Let stand for a moment or two, then skim as much fat as possible from the surface of the sauce. Taste for seasoning and add salt and pepper, as desired. In the meantime, quickly saute the sliced mushrooms in the butter until lightly browned and tender. Add the mushrooms to the sauce and serve with the roast, carrots and potatoes. Rather than carve the meat you can simply pull chunks apart and plate them with the potatoes and carrots. If you prefer, cook the potatoes separately and mash them. Set the meat and carrots atop the mash then add the mushrooms and sauce. Sprinkle chopped fresh herbs over as a garnish. If preparing the roast ahead of time, cover it and cool slightly, then wrap tightly and refrigerate for up to three days. Package the sauce and vegetables separately and refrigerate. When ready to serve, lift any fat from the sauce, add the meat and vegetables, and heat in a low oven.