Add peaches, mint leaves, and simple syrup to the bottom of a cocktail shaker. Gently muddle to release the peach juices and mint oils. (Over-muddling the mint will cause it to become bitter.)
Add bourbon, fill the shaker with ice, and shake until well-chilled. Strain into a rocks glass filled with fresh ice.
Top with ginger beer or seltzer water. Garnish with fresh mint sprigs and a peach slice.
*We prefer a nice spicy ginger beer to finish this cocktail. Note that if you're using seltzer water, you might enjoy adding a little lemon juice (about 1/2 ounce) to the shaker to balance the flavors.
About the Brown Sugar Simple Syrup:
The quantity of syrup you need will depend on the sweetness/juiciness of your peaches, the bourbon you're using, and your personal tastes. We typically make this cocktail with 1 ounce of simple syrup, but when I've gotten very sweet peaches at the market, I've reduced it to 1/2 to 3/4 ounce.To make Brown Sugar Simple Syrup, combine equal parts brown sugar and hot water until sugar is dissolved. Cool to room temperature and store, covered, in the refrigerator for up to 3 weeks.
Omit bourbon and simple syrup. Muddle the peaches and mint with 1-1/2 ounces of peach juice and 1/2 ounce of lemon juice. Strain over ice, and top with ginger beer or ginger ale. Add simple syrup only to taste.