4 ripe unpeeled peaches, pitted and coarsely chopped
1 1/2 cups white sugar
1 cup water
3 sprigs basil, tied together with string
ice cubes, divided
1 1/2 fluid ounces bourbon
3 dashes aromatic bitters (such as 18.21 Prohibition Aromatic Bitters)
2 fluid ounces lime-flavored sparkling water (such as La Croix®)
1 slice peach
1 leaf basil
Combine chopped peaches, sugar, water, and 3 sprigs basil in a small pot over medium heat. Simmer, stirring occasionally, until sugar dissolves, about 5 minutes. Cool syrup to room temperature, about 10 minutes. Discard basil sprigs.
Pour syrup into a Mason jar, seal, and refrigerate until completely chilled, about 1 hour.
Measure 3 fluid ounces of syrup and peaches into a cocktail shaker; muddle to break peaches up. Add 4 ice cubes, bourbon, and bitters; cover and shake vigorously until ice is mostly melted. Pour into a chilled 8-ounce glass; add remaining 4 ice cubes. Gently stir in sparkling water. Garnish with a peach slice and basil leaf.