Place the balsamic vinegar and sugar in a small sauce pan. Bring to a boil then reduce heat to medium low and let the mixture simmer while you continue on with the recipe. Once the vinegar is thickened and syrupy, remove from heat.
Meanwhile, heat the remaining olive oil in a large skillet over medium heat. Season the diced chicken breast with salt and pepper then place them in the skillet. Cook for a few minutes on each side until the pieces are cooked through.
Preheat the broiler to low. Split the sub buns in half and brush them lightly with about 1 Tablespoon of olive oil. Place them oil side up on a baking sheet and place them under the broiler for 1-2 minutes or until golden. Remove and set aside.
Spread the chicken evenly over the toasted bread. Top with slices of mozzarella cheese. Place the cheesy bread under the broiler for an additional 1-2 minutes or until the cheese is melted and bubbly.
While the cheese is melting, combine the tomatoes and garlic and season them with salt and pepper. Once the cheese is melted, spread the tomatoes out evenly over the bread slices. If desired, place them back under the broiler for a final time to warm up the tomatoes.
Top bread slices with chopped basil and a drizzle of the reduced balsamic vinegar. Serve immediately.