Make the Peach Syrup:
In a medium saucepan, combine granulated sugar and water over medium heat. Stir constantly until the sugar dissolves, about 3–5 minutes. Add half of the sliced peaches (about 3 peaches) to the saucepan, bring to a gentle simmer, and cook for 10–15 minutes, stirring occasionally until the peaches soften and release their flavors.
Remove from heat and strain the mixture into a heat-resistant bowl using a fine-mesh sieve. Press the peaches to extract all the juice. Let the syrup cool completely.

Blend the Sorbet Base:
Once the peach syrup has cooled, combine it in a blender or food processor with the remaining sliced peaches, lemon juice, and peach schnapps (if using).
Blend until smooth, scraping down the sides as needed. Taste and adjust the sweetness with extra sugar or lemon juice if needed. Keep in mind that the flavors will be milder once frozen.

Chill the Mixture:
Pour the sorbet base into a container with a lid. Cover and refrigerate for at least 4 hours, or preferably overnight, to allow the flavors to meld and ensure the base is fully chilled before churning.

Churn the Sorbet:
Remove the ice cream maker bowl from the freezer and assemble the ice cream maker according to the manufacturer’s instructions.
Pour the chilled sorbet base into the ice cream maker and churn for 20–30 minutes, or until the sorbet reaches a soft-serve consistency.

Freeze the Sorbet:
Transfer the churned sorbet into an airtight container and freeze for at least 2 hours, or up to 4 hours, to firm it up before serving.

Serve:
Let the sorbet sit at room temperature for 5–10 minutes before serving, which will make it easier to scoop. Serve in bowls or cones, and garnish with fresh peach slices, mint leaves, or a drizzle of honey for an extra special touch.