Set a grill to low heat and preheat for 10 minutes or so, brush the grill grates with a paper towel dabbed in oil. I use grapeseed oil.
Wash and dry peaches. Cut each peach in half and then each half into 4 quarters. Place one large basil leaf on each side of the peaches. Cut the bacon slices in half. Wrap each peach slice and basil leaves with a slice of bacon. Pin the loose end of the bacon slice with a toothpick. Repeat with remaining peaches.
Brush the bacon wrapped peaches lightly with olive oil so the bacon doesn't stick to the grill. Grill the peaches until the bacon is cooked, turning so all sides are evenly cooked, about 20 minutes.
Transfer to a serving platter and remove toothpicks. Drizzle with store-bought or homemade balsamic glaze.