Prepare the grill for direct cooking over low heat (250° to 350°F).
Brush the cooking grates clean. Brush the bread slices on both sides with the oil. Grill the bread on one side over direct low heat, with the lid closed, until toasted, 1 to 2 minutes. Transfer the bread to a work surface with the toasted sides facing up. Rub the toasted sides vigorously with the garlic clove. Place a slice of cheese on top of each. Grill over direct low heat, with the lid closed, until the cheese melts slightly, 1 to 2 minutes. Transfer the bread to a plate.
In a small bowl toss the chopped tomatoes with the oil and vinegar and season with salt and pepper. Top the toasted bread slices with the tomato mixture and garnish with basil. Serve warm or at room temperature.
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