Add sliced peaches, honey, 1 oz lemon juice, and basil stem to a small pan. Bring to a boil over medium heat, then lower heat to a simmer for 10-15 minutes.
Remove peaches from heat, remove basil stem, and mash peaches. Cool in the refrigerator for 10-15 minutes.
Divide peach mash into 4 serving glasses. Top with ice, rye, peach schnapps, remaining lemon juice, and sparkling water. Garnish with basil leaves and peach slices, if desired.
Notes
If you want even more peach flavor, swap out 2 oz of the rye with 2 oz of peach schnapps. You can also muddle up some roughly chopped basil in the bottom of your glass for a more intense basil flavor.