In a small nonaluminum saucepan, bring the vinegar to a simmer over medium-low heat. Cook until reduced by half, about 5 minutes. Remove from the heat and let cool.
Place the halved strawberries in a blender or food processor. Process until very smooth. Pass the puree through a fine-mesh sieve placed over a bowl, pressing firmly on the solids with a rubber spatula or the back of a wooden spoon to extract all the juice. Discard the solids. Add the balsamic reduction and the honey to the puree and stir to combine. Cover and refrigerate until cold.
Freeze the strawberry mixture in an ice-cream maker according to the manufacturer's instructions. Store in the freezer until ready to serve or for up to 2 days. Spoon into individual bowls and garnish with the chopped strawberries.