Rub the lemon wedge on half of the outside of a coupe glass, coat with the basil salt mixture and set aside.
Add the gin and syrup to a cocktail shaker with ice and shake until chilled.
Double-strain into the prepared coupe glass.
Float the chilled prosecco on top.
*Honeydew-basil-white-balsamic shrub: Cut off the rind and remove the seeds from 1 whole honeydew, then cut the melon into chunks. Combine it in the blender with 1 cup fresh basil, and blend until smooth. Pour the mixture into a saucepan, add 4 cups white sugar and 4 cups cold water, and simmer until the sugar has dissolved, being careful not to overheat. Remove it from the heat and let it cool, then strain through a chinois cloth. Add 1 ½ cups white balsamic vinegar, and keep the shrub in the refrigerator for up to two weeks.