1. Preheat the oven to 400°. Cut four pieces of parchment paper, each 15-inches-by-20-inches. Brush each with a bit of olive oil. On each parchment rectangle, place ¼ cup cherries, ¼ cup tomatoes and 2 basil leaves. Lay a fillet over each mound of vegetables. Season each with salt and pepper and drizzle with a bit of olive oil and a tablespoon of vinegar.
2. To seal the packages, fold up the two longer sides of the parchment paper toward each other, laying one side over the other. Fold the shorter ends of the package together and then tuck them under the rest of the package. Repeat with all four packages, ensuring that each is securely closed.
3. Place the parchment packets on a sheet pan and place in the oven. Cook for 10 to 15 minutes, until the fish is opaque and warmed through. Serve immediately, allowing diners to open the packets at the table.
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