Preheat the oven to 275° F.
Arrange the tomatoes on a sheet pan, cut sides up, in a single layer.
Drizzle with 1/4 cup of olive oil and the balsamic vinegar. Sprinkle with the garlic, sugar, 1 1/2 teaspoons salt, and 1/2 teaspoon pepper. Roast for 2 hours until the tomatoes are concentrated and begin to caramelize.
Allow the tomatoes to cool to room temperature.
Cut the mozzarella into slices slightly less than 1/2-inch thick. (If the slices of mozzarella are larger than the tomatoes, cut the mozzarella slices in half.)
Layer the tomatoes alternately with the mozzarella on a platter and scatter the basil on top.
Sprinkle lightly with salt and pepper and drizzle lightly with olive oil and a very good quality, thick balsamic vinegar or syrup.
Serve at room temperature.