If desired, place peach quarters and lemon, cut sides down, on the greased rack of a covered grill directly over medium heat. Grill 5 minutes or until browned, turning peach quarters once.
In a cocktail shaker combine peach quarters, lemon half, basil, and honey. Use a muddler or the back of a spoon to crush the fruit and release juices. Add bourbon and ice cubes; cover and shake until very cold. Strain into an ice-filled glass. Top with club soda.