Heat the oil in a large oven-safe casserole over medium-high heat. Add the mushrooms and saute until brown, 3 to 5 minutes. Add the salad-dressing mix and onions and toss to coat evenly. Stir in the wine and broth and cook until blended. Add the chicken. Cover, reduce heat to low, and simmer 15 minutes, turning pieces halfway through. Remove the chicken and vegetables. Increase heat to high and boil the sauce until reduced and slightly thickened. Pour over the chicken.