Place the potatoes in a large pot, cover with cold water, and bring to a boil.
Add 1 ½ tablespoons salt, reduce heat, and simmer until tender, 12 to 15 minutes. Drain, run under cold water to cool, and slice.
Meanwhile, in a small bowl, whisk together the oil, vinegar, mustard, ½ teaspoon salt, and ¼ teaspoon pepper.
In individual bowls, combine the arugula, chicken, tarragon, and potatoes. Drizzle with the dressing and sprinkle with the Parmesan.