For the Peach Shrub Syrup: in a pot on the stovetop, combine a cup of fresh ripe peaches, pitted, chopped, and muddled; a cup of white balsamic vinegar, and a cup of granulated sugar (we used turbinado).
Bring to a very brief boil, then simmer for 10-20 minutes or longer, stirring frequently and tasting occasionally.
Once the sugar has melted and the vinegar has absorbed lots and lots of peach flavor, remove from the heat and strain well – probably two or three times, through a sieve or cheesecloth – and let it cool.
Store it in a bottle in the refrigerator for a few weeks to a month.