Hull and roughly chop strawberries. In a bowl, combine the strawberries with 1/3 cup honey and 1 teaspoon balsamic vinegar, and let stand for 15 minutes.
Blend together the strawberries with 1 cup Greek yogurt and 10 basil leaves using a blender or immersion blender, until thoroughly combined.
Pour the mixture into popsicle molds and freeze until firm, about 4 hours.