Cut each tomato in half and place in a large bowl. Sprinkle with sugar and a large pinch of salt. Toss to combine and let sit for 30 minutes.
Place the tomatoes and liquid in a salad spinner. Spin to remove the seeds and any excess liquid. Place the tomatoes back into the large bowl and pour the liquid through a fine mesh strainer into a small saucepan. There should be about 1/3-1/2 cup liquid.
Place the saucepan over medium heat and stir in the vinegar and garlic. Bring to a simmer and cook until the liquid reduces to about 3 tablespoons. Remove from heat and let sit until room temperature.
Whisk the olive oil into the balsamic mixture once cooled. Pat any excess moisture from the mozzarella and toss with the tomatoes, along with the basil. Coat with the cooled dressing, season with salt and pepper, and serve.