Prepare the syrup: In a small saucepan, combine sugar and water. Heat over medium heat, stirring until the sugar dissolves completely. Remove from heat and let it cool.

Blend the berries: In a blender, combine the mixed berries, lemon juice, and vanilla extract. Blend until smooth.

Strain the mixture: Pour the blended berry mixture through a fine-mesh sieve to remove seeds. Press with a spoon to extract as much liquid as possible.

Combine with syrup: Add the cooled sugar syrup to the strained berry mixture. Stir well to combine.

Chill the mixture: Pour the mixture into a shallow dish and place it in the freezer for about 1 hour, until it begins to freeze around the edges.

Whisk and freeze: Remove from the freezer and whisk vigorously to break up any ice crystals. Return to the freezer and repeat this process every 30 minutes for about 2-3 hours, until the sorbet is firm but still scoopable.

Serve: Scoop the sorbet into bowls or cones and enjoy immediately. For a firmer texture, freeze for an additional hour before serving.