Preheat the oven to 275 degrees 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. Serve at room temperature.
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