Mix chicken with tomatoes, 2 tablespoons basil, balsamic vinegar and olive oil. Season to taste with salt and pepper. Mixture can be kept, covered, 2 hours in refrigerator. Bring to room temperature.
Lightly brush bread with olive oil. Broil it until lightly toasted. Rub toasted sides with cut garlic.
Top 2 slices of the toast with chicken mixture and let stand 5 minutes to soften slightly. Sprinkle with remaining basil, top with remaining toasts and serve.
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