Put everything except the beetroot into a food processor and run until slushy. (You will probably want to split the tomatoes into 2 batches to avoid spillage.).
Place 4 layers of clean muslin cheesecloth in a deep bowl. Pour the tomato mixture into the cloth. Tie up the corners of the fabric. Add the slice of beetroot to the bowl to color the liquid. Hang the bag from a shelf in the refrigerator with the bowl underneath for 6-8 hours (or longer). Discard the beetroot.
Serve in a pretty clear bowl with an ice cube to keep it very cold, a nice basil leaf, and a few drops of very good extra-virgin olive oil.