Core, peel and seed tomatoes; cut flesh into 1-inch chunks. Seed melon, and remove flesh from rind; cut into chunks. Place a tablespoon of olive oil in each of two 10- or 12-inch skillets and turn heat under both to high. Add melon to one and tomatoes to the other, and cook, stirring, until they become juicy, no longer than 2 minutes.

In a blender or food processor, puree melon with tomatoes, 1 ½ cups water and basil, along with some salt and pepper. Stir in remaining olive oil. Chill, add lemon juice to taste and adjust seasoning. Serve.