Place the potatoes in a large pot, cover with cold water, and bring to a boil. Add 1 1/2 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, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Using a fork or your fingers, shred the chicken, discarding the skin and bones.
In individual bowls, combine the arugula, chicken, tarragon, and potatoes. Drizzle with the dressing and sprinkle with the Parmesan.
Make Ahead: You can cook the potatoes, shred the chicken, and make the vinaigrette up to 24 hours in advance. Just before serving, slice the potatoes and assemble the salad as directed.