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.