Place potatoes in a large pot, and cover with cold water; bring to a boil. Add 1 tablespoon salt, reduce heat, and simmer 15–18 minutes or until tender. Drain and run under cold water to cool; peel and slice.
Fill a medium saucepan halfway with water. Bring to a boil, and add remaining 1 tablespoon salt. Add chicken, and reduce heat to medium; gently simmer 12–14 minutes or until cooked through. Transfer to a cutting board, and let rest at least 10 minutes before shredding with a fork or your fingers.
Whisk together oil, vinegar, mustard, and pepper in a small bowl.
Divide arugula, chicken, tarragon, and potatoes evenly into 4 serving bowls. Drizzle each with dressing, and top with Parmesan.