In a bowl, whisk 1 1/2 tablespoons of the sugar with 1/2 tablespoon of the lemon juice until the sugar is dissolved. Whisk in the buttermilk and pour into a shallow baking dish; freeze until firm, whisking the mixture every 30 minutes, about 3 hours.
In a bowl, toss the strawberries with the remaining 2 tablespoons of sugar and 1 tablespoon of lemon juice. Add the lemon zest and vinegar; let stand for 30 minutes. Spoon the berries and any juices into glasses. Using a fork, scrape the buttermilk ice into fluffy crystals and spoon over the strawberries. Garnish with the tarragon and serve.