Place the tomatoes and mozzarella in a large bowl. Put the garlic, mustard, vinegar, 1/2 teaspoon salt, and 1/4 teaspoon pepper in a 1-cup glass measuring cup and slowly whisk in the 1/2 cup of olive oil. Set aside.
Heat the remaining 1/3 cup of olive oil in a large (12-inch) saute pan until hot but not smoking. Add the bread cubes and sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper. Saute over medium to medium-high heat for 5 to 8 minutes, tossing occasionally, until the bread is evenly browned and crisp. Add the warm bread to the tomatoes and mozzarella. Add enough of the vinaigrette to moisten all the ingredients. Add the basil, sprinkle with salt and pepper, and toss carefully, adding more vinaigrette if necessary. Serve at room temperature.