Peel the potatoes. If they are not very small, cut them in half. Put them in a saucepan and cover with water.
Trim and peel the carrot. Cut into 2-in. length, then cut into quarters. Peel the onions and cut them in half. Add the carrots and onions to the potatoes. Add salt to taste and bring to a boil. Let stand 10 min.
Meanwhile, sprinkle the chicken halves with salt and pepper. Brush on all sides with mustard. Arrange the halves side by side in a baking dish, skin side down. Sprinkle with the oil. Place under the broiler and cook for 10 min.
Drain the vegetables. Melt the butter in a skillet and when quite hot but not brown, add the vegetables. Cook over relativly high heat, shaking the skillet and stirring the vegetables.
Add the mushrooms and sprinkle with salt and pepper to taste. Continue cooking, shaking the skillet and stirring occasionally, about 7-8 min.
Pour off all the fat from the baking dish and turn chicken halves again.
Turn then oven heat to 205C. Place the dish in the oven and bake 10 min.
Place the dish on top of the stove. Add the vinegar to the liquid and cook for a few seconds over high heat,stirring. Sprinkle with parsley.