Center a rack in the oven and preheat it to 450˚F. Rub a baking sheet with a little oil (or line it with foil and oil the foil).
Put the potatoes, mushrooms, shallot or onion and garlic in a large bowl. Toss in 4 sprigs each of the rosemary and thyme, 1 teaspoon salt, a few grindings of pepper, 3 tablespoons of the oil and 3 tablespoons of the balsamic. Mix everything well and spread the ingredients out on the baking sheet.
Put the chicken in the bowl, along with the remaining 3 tablespoons oil and 2 tablespoons balsamic, 1/2 teaspoon salt and a good amount of pepper, and mix well to coat the chicken. Move the vegetables around to make room for the chicken. Tuck the remaining herbs under each piece.
Roast the chicken for 40 minutes to 1 hour, until a thermometer stuck into the thickest part of a thigh measures 165˚F.
Serve everything on the baking sheet or a big platter; pour over the cooking juices.