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Core, peel, and seed the tomatoes; cut the flesh into 1-inch chunks. Seed the melon, and remove the flesh from the rind; cut into chunks. Place a tablespoon of olive oil in each of two 10- or 12-inch skillets and turn the heat under both to high (you can do this sequentially if you prefer).
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Add the melon to one and the tomatoes to the other and cook, stirring, until the become juicy, no longer than 2 minutes.
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Blend together the melon, tomato, water, and basil, along with some salt and pepper. Chill, then add lemon juice to taste and adjust seasoning.