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Preheat oven to 400ºF with rack in middle.
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Heat oil in a 10-inch heavy skillet over medium-high heat until it shimmers, then sauté garlic until pale golden, about 30 seconds. Add tomatoes and a pinch of salt and sauté until tomatoes are softened, about 1 minute. Stir in capers. Remove from heat.
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Cut fish into quarters (you are aiming for square-shaped pieces rather than long, thin cross-cut slices). Pat dry and sprinkle both sides with 3/4 teaspoon salt and 1/4 teaspoon pepper (total).
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Arrange 4 (12- to 15-inch) squares of parchment paper on a work surface. Drizzle center of each parchment square with a little olive oil, then top with a fish fillet. Slide lemon slices under fish and top with thyme sprigs. Spoon tomato mixture over fish.
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Gather parchment up around fish to form a pouch, leaving no openings, and tie tightly with string (make a bow so that you can untie it easily). Put pouches in a large shallow baking pan and bake until fish is just cooked through, about 18 to 20 minutes. (Check by unwrapping one pouch.)