Preheat grill to medium. In a small bowl, combine tomatoes, mozzarella, 1 teaspoon olive oil, oregano, pepper, and garlic. In a second small bowl, whisk remaining olive oil, lemon juice, and vinegar.

Using a pastry brush, brush oil-lemon juice mixture on both sides of mushroom caps. Grill, stemside down, for 5 minutes or until soft.

Spoon tomato mixture into each cap, replace on grill, cover, and grill 3 minutes more, or until cheese melts. Sprinkle with fresh parsley or basil. Salt to taste and serve hot.