Sauce: In small saucepan, whisk together vinegar, tomato juice, oil, thyme, mustard, sugar and garlic; bring to a boil over high heat. Boil, whisking occasionally, for 5 minutes, until thick enough to coat back of spoon. Set ¼ cup aside to serve with chicken.
Halve tomatoes lengthwise. Remove mushroom stems. Arrange vegetables, cut side up in a 13 X 9 inch cake pan. Spoon half of the sauce over the mushrooms only. Sprinkle both vegetables with cheese. Bake on lower rack in 400oF oven for 10 minutes.
If you bought chicken with skin, remove it now. Score meat three times. Coat with some of the remaining sauce. Broil, turning and brushing with sauce once, for 12 – 15 minutes or until no longer pink inside. Season chicken and vegetables with salt and pepper to taste. Serve with reserved sauce.
The only change I made to this recipe was to decrease the olive oil used from 3 tbsp to 1 tbsp. You might want to fiddle with that amount, taking your tolerance for fats into consideration.