Make the ice cream base according to the Classic Ice Cream Method.
Strain the base through a fine-mesh strainer and cool over an ice bath. Age the base under refrigeration overnight.
Once the base has aged, add the balsamic vinegar to the base right before freezing (in the case of a Pacojet) and right before churning (in the case of a batch freezer). Otherwise, its acidity will curdle the milk and heavy cream. Churn to the desired consistency.
Harden for at least 2 hours and up to 4 hours. Reserve until needed.