Using a fork or your fingers, shred the chicken meat, discarding the skin and bones. In a large bowl, combine the chicken, carrots, scallions, tarragon, 1/2 teaspoon salt, and 1/4 teaspoon pepper.
In a small bowl, whisk together the mustard, vinegar, 1/4 teaspoon salt, and 1/4 teaspoon pepper. Slowly add the oil in a steady stream, whisking constantly until emulsified. Divide the radicchio leaves among individual plates, spoon the chicken salad over the top, and drizzle with the vinaigrette.
Substitution: If you find radicchio too peppery, spoon the salad into Boston or Bibb lettuce leaves. Or serve it in endive leaves, as fork-free hors d'oeuvres.