Place the tomatoes and basil into a large pot or bowl. Use a stick blender, food processor, or blender to puree the tomatoes and basil together. Gradually blend in the oil to form a good emulsion. If pureeing the mixture in smaller batches, add a bit of oil to each one. Season with salt and ground black pepper, but remember, the Parmesan ice cream will be salty. Place the tomato preparation in the refrigerator while the ice cream is prepared.

Heat the cream in a small saucepan over medium heat. Stir in the Parmesan cheese, and continue to cook and stir until the cheese is melted, and the mixture is uniform. Season with a dash of freshly ground black pepper, and remove to a small bowl to cool. This will temper the mixture, so it will freeze more smoothly and evenly. When the cheese mixture has cooled a bit, place it in the freezer.

Take the tomato mixture from the refrigerator, and pass it through a chinois, food mill or strainer with medium holes to remove bits of peel and seeds. The result should be a creamy liquid, not a tomato juice.

Serve the cold soup in small bowls or large cups. Make little balls with the frozen cheese using a dessert spoon. Place a little ball of the Parmesan ice cream into each bowl just before serving.