Cut carrots into long, thin strips using a vegetable peeler. Combine carrot, spinach, and next 3 ingredients in a large bowl. Set aside.
Combine tarragon and next 7 ingredients in a small jar; cover tightly, and shake vigorously.
Add vinegar mixture to spinach mixture, and toss well. Place spinach mixture evenly on 4 plates. Top evenly with chicken. Serve with Goat Cheese Crostini.