Freeze ice cream bowl. Place the bowl of an ice cream maker in the freezer 24 hours prior to churning.
Make basil simple syrup. Combine the sugar and water in a small saucepan over medium heat. Heat until the sugar dissolves, stirring occasionally.
Add basil. Add the basil leaves and reduce heat to a gentle simmer. Let steep for 10 minutes, stirring occasionally. Strain out the basil and let the simple syrup cool to room temperature.
Blend. Add the thawed chilled berries to a high-speed blender with lemon juice and cooled simple syrup. Blend until smooth. Then run the mixture through a fine mesh sieve (for seedless sorbet).
Chill. Chill the pureed mixture for at least 2 hours.
Churn. Pour the sorbet mixture into the ice cream maker bowl and process according to your machine's instructions until thick, typically around 15-20 minutes.
Freeze. Spoon the sorbet into a freezer-safe container and cover with a lid or tightly with plastic wrap and freeze until solid, at least 6 hours. Take the sorbet out of the freezer a few minutes before scooping.