Heat a grill or grill pan to medium-high. Brush the cut sides of the peach halves with oil and sprinkle them with the brown sugar.
Grill cut-side down for 4 to 5 minutes without moving them, until the flesh is deeply marked and the sugar has caramelized. Turn and give the skin side 2 minutes more.
Let the peaches cool for 10 minutes, then slip off the skins — they come away easily once grilled — and roughly chop the flesh.
Whisk the honey into the hot water until it dissolves completely. Honey seizes into a lump in cold liquid, so this step is not one to skip.
Put 4 of the chopped grilled peaches into a large pitcher with the basil leaves and the salt. Muddle firmly, pressing and twisting, until the fruit breaks down and the basil is bruised but not shredded.
Add the bourbon, lemon juice, 3/4 cup of honey syrup, and the remaining chopped peaches. Stir well.
Cover and refrigerate for at least 2 hours, or up to 12. Stir once partway through.
Strain into a serving pitcher if you prefer a clean drink, or leave the fruit in for a rustic one. Add the club soda just before serving, pour over ice, and garnish each glass with a basil sprig and a peach wedge.