Fold a full sheet of paper towel into quarters and place it in a small bowl. Spoon the yogurt on top of the paper towel. Place it in the refrigerator for 20 minutes so the yogurt can drain and thicken.
In a small bowl whisk together the thickened yogurt and the mayonnaise, until smooth. Whisk in the lemon juice, vinegar, and honey.
Grate first the carrots and then the apples in a food processor. Transfer them to a large bowl and stir to combine. Pour the dressing over the carrot mixture and toss. Add the mint and season with salt.