Whisk the ayran thoroughly with honey and cornstarch until smooth.

Heat the mixture in a small pot over low heat, stirring constantly, until slightly thickened, then remove from heat without letting it boil.

Once the mixture has cooled to room temperature, add the mint and stir, then transfer to a shallow dish.

Place the dish in the freezer and stir with a fork every hour for the first 3-4 hours, then freeze covered for at least 1 hour more.

A few minutes before serving, remove from the freezer, scoop into balls with a spoon, and serve in chilled glasses.