Chop up tomatoes and process them in a blender.
Press the tomato pulp through a chinois or fine mesh strainer. Once you have most of the solid matter separated out, allow the juices to run through several layers of cheesecloth for several hours, resting the temptation to press the liquid through the cloth. This will produce a perfectly clear tomato water. You will want about 2 to 3 cups of the water.
While the tomato water is dripping through the cheesecloth, make the simple syrup. In a pot, bring to a boil 2 cups water and 2 cups sugar. Allow it to simmer until all the sugar is dissolved. Add a fist full of fresh basil leaves and allow the syrup to cool completely. This can be made several days ahead. The longer you allow the basil to steep in the syrup the stronger the flavor will be.
Once the tomato water is done straining, add enough of the simple syrup to make an egg float. Yes, you read that right. Follow these directions for making smooth sorbet. Depending on the variety of tomatoes you use, the sorbet may seem too sweet, so you can squeeze half a lemon into the mix and retest with the egg. The sorbet is also lovely with some basil leaves finely chopped and added to the mix. Freeze the tomato sorbet as your ice cream machine suggests.
