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.