Combine water and sugar in a large saucepan over medium-high heat. Bring to a boil, whisking often. Reduce heat to medium-low and simmer for 5 minutes, continuing to whisk, until sugar is dissolved.
Add basil leaves to syrup and cool to room temperature. Once mixture has cooked, transfer to a large bowl and cover. Place syrup in refrigerator and refrigerate for at least 2 hours.
Strain syrup through sieve to remove basil; squeeze out remaining liquid from basil leaves and discard. Whisk lemon juice and zest into basil syrup.
Add lemon-basil mixture to ice cream maker and freeze according to manufacturer's instructions. Once sorbet is solid, transfer to an airtight container and freeze for at least 2 hours before serving.
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