In a medium saucepan, whisk together the cream and milk until steaming.
Meanwhile, whisk together the yolks, sugar and vanilla.
Gradually whisk some of the hot cream mixture into the yolk mixture to heat it up; return the mixture to the saucepan. Stir over medium heat until the custard thickens; strain through a sieve set over a bowl. Cover and refrigerate until well chilled.
Meanwhile, simmer the balsamic vinegar in a small saucepan until reduced to 2 tablespoons, about 6 minutes. Cool syrup completely. (Alternatively, start with some balsamic reduction and skip this step.)
Freeze the custard in an ice cream maker according to manufacturer's instructions. When ice cream is done, spoon in balsamic syrup and churn 3 to 4 seconds longer to swirl.
Serve immediately or transfer to a container to freeze until solid.