Start a charcoal or wood fire or heat a gas grill or broiler. The fire should not be too hot, but the rack should be fairly close to the heat source, 4 inches or less. If necessary, you can pound the chicken slices lightly between 2 pieces of waxed paper so that they are of uniform thickness.
Rub the chicken with 1 tablespoon olive oil. While the grill is heating, heat remaining 2 tablespoons olive oil in a medium to large skillet over medium heat. Add garlic and cook, stirring, until it colors lightly. Add capers and cook for 15 seconds. Add tomatoes and cook over medium-high heat until thick, about 10 minutes. Taste and add salt and pepper to taste, then cook 1 minute more. Set aside, keeping warm.
Grill or broil the chicken very quickly; it should take no more than 3 to 4 minutes per side. Serve the chicken with the tomato sauce, garnished with minced fresh basil or parsley leaves.
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