Preheat the oven to 400 degrees F. Toss the quartered tomatoes with a large pinch of salt, some freshly ground pepper, and 1 tablespoon of olive oil. Spread tomatoes in a single layer on a small baking sheet.
Roast tomatoes for 30-40 minutes until soft and beginning to brown. Turn on the broiler for a couple of minutes to further caramelize the edges.
While the tomatoes are roasting, cook the chicken and make the balsamic reduction. Pour the balsamic vinegar into a small saucepan and whisk in the sugar. Bring to a boil, reduce heat to medium-low and simmer until thickened slightly and reduced by about half. About 5 minutes.
Sprinkle chicken with salt and pepper on both sides. Heat a tablespoon of olive oil in a large non-stick skillet over medium-high heat. Sear the chicken on one side until lightly golden, about 2 minutes. Turn breast over, reduce heat to low and cover with a snug-fitting lid. Let cook for about 15 minutes.
Put mozzarella slices on top of the chicken, and add the roasted tomatoes to the pan. Cover and let sit on low heat for a couple of minutes, until cheese is melted, or if your skillet is oven-safe, pop it under the broiler for a minute.
Drizzle with balsamic reduction, and sprinkle with chopped, fresh basil.Serve with buttered noodles, or a green salad.