In a small saucepan, bring vinegar to a boil; reduce and cook until liquid is reduced to 3 tablespoons, 5-6 minutes. Remove from heat. Meanwhile, combine tomatoes, shallot, basil, 2 teaspoons olive oil and garlic. Cover and refrigerate until serving.
Brush remaining oil over both sides of baguette slices. A spray oil works well. Grill, uncovered, over medium heat until golden brown on both sides. I had the grill running at 500 degrees, and cooked each side for 1 minute.
If not grilling, broil slices on low in two minute intervals, turning after each interval until toasted. May take 6-10 minutes
Top toasts with tomato mixture. Drizzle with balsamic syrup; sprinkle with sea salt and Parmesan. Cook for an additional minute or two. Serve immediately.
