12-16ozcherriesfresh or frozen and slightly thawed, pitted
1tablespoonlemon or lime juice
Combine water and sugar in a small saucepan and bring to a boil over medium-high heat. Reduce heat to low and simmer until sugar dissolves, stirring periodically, about 3-4 minutes. Remove the simple syrup from the heat and drop in your basil, using more or less depending on how pronounced of a basil flavor you wish. Place in the refrigerator to cool for 1 hour. Remove and discard basil. You can use the basil simple syrup as soon as it is cool or store it in a tightly sealed glass jar until you are ready to use, up to several days in advance.
Place cherries, 1 cup of basil simple syrup, and lemon juice in a blender. Adjust sweetness as necessary with remaining simple syrup, depending on the sweetness of the cherries. Blend and strain. Chill for at least 2 hours or overnight. The sorbet mixture can be made up 24 hours in advance.
Put mixture into an ice cream maker and churn for 20 to 30 minutes. Transfer to and store in an airtight container in the freezer to firm for at least 5 hours, ideally overnight.