Slice chicken breasts in half width-wise (to create 4 total thin chicken breast halves. Place them in a baking dish or plastic bag with the Italian dressing. Let sit for 30 minutes to 6 hours.
Wrap the sliced baguette in foil.
Pour the balsamic vinegar into a small saucepan and heat over medium. Let simmer (just small bubbles) for 5 minutes, then turn off the heat and let thicken for an additional 10 minutes with the stove off.
In a small bowl, mix together the diced tomato, torn basil, olive oil, salt and pepper. Let sit for 10 minutes.
In the meantime, heat the grill and add the foil-wrapped bread to the top rack to warm up. Place the chicken breasts on the grill under the direct heat with tongs. Cook for about 5 minutes per side, until cooked through.
Add the diced mozzarella to the tomato mixture and toss.
Plate the chicken breasts on one side of a platter. Add the toasted bread to the other side. Sprinkle with the bruschetta tomatoes, then drizzle with the balsamic glaze. Top with additional basil leaves