Prepare the simple syrup: In a small saucepan, combine the water and sugar. Heat over medium heat, stirring until the sugar is completely dissolved. Do not boil excessively. Remove from heat and allow to cool completely.

Puree the peaches: Place the chopped peaches and lemon juice into a high-speed blender or food processor. Blend until perfectly smooth.

Combine the base: Strain the peach puree through a fine-mesh sieve into a bowl to remove any remaining pulp or fibers, pressing down on the solids. Whisk in the cooled simple syrup and the optional peach liqueur.

Chill thoroughly: Cover the peach mixture and refrigerate for at least 2 hours, or until completely cold. A colder mixture churns better and yields a better texture.

Churn the sorbet: Pour the chilled mixture into your ice cream maker or sorbet maker, following the manufacturer's instructions. Churn until the sorbet reaches a soft-serve consistency (usually 20-30 minutes).

Harden the sorbet: Transfer the churned sorbet to an airtight, freezer-safe container. Freeze for an additional 2 to 4 hours to allow it to firm up completely before serving. Let stand at room temperature for 5-10 minutes before scooping.