Using a juicer, run the peaches through the juice extractor. You'll get about 1 cup of peach juice. Or place the peaches in a blender -- just keep in mind it will be more of a smoothie consistency if you don't juice them, but that's okay too!
Finely chop basil or muddle it to help release the natural oils because that's where all the flavour is.
Divide peach juice and basil evenly amongst 4 champagne glasses. Fill glass with fizzy water. Enjoy immediately!