Preheat oven to 350 degrees. Cut squash in half, lengthwise. Remove seeds from inside. Drizzle olive oil over the inside, and rub to cover all flesh. Add dash of salt and pepper. Place face down on a baking sheet/dish. Bake for 45-60 minutes or until tender (time will vary depending on size of squash). Remove, let cool for at least 15 minutes, then using a fork scrape out the squash, while being careful not to poke holes in the skin. Set aside.
Combine balsamic vinegar and sugar in a small pot. Bring to a boil over medium heat, then simmer for about 5-8 minutes or until thickened. Do not overcook or it will harden.
Preheat oven to 350 degrees.
Cut fresh mozzarella ball in half. Using one half, cut into small cubes. With the remaining half, slice into thin rounds, set aside.
In a large bowl, add spaghetti squash filling, mozzarella cubes and 1/4 cup shredded mozzarella, chopped tomatoes (reserve rounds), basil leaves salt, and pepper. Mix together. Add about 2 TBS of the balsamic glaze, stir to combine.
Scoop the mixture into both sides of the skin of the spaghetti squash. Top with mozzarella rounds and tomato rounds. Loosely cover with foil and bake for 15 minutes. Remove foil, turn oven to broil, and broil for 3-5 minutes, or until mozzarella is slightly browned (watch carefully as it can burn quickly). Remove from oven.
Drizzle remaining glaze over the tops, serve immediately.