The chicken and potatoes roast first together. Mushrooms are added later and then stirred so that they roast evenly.
Preheat oven to 500°.
Remove and discard giblets and neck from the chicken. Rinse chicken under cold water, and pat dry. Trim excess fat from chicken. Place chicken, breast side up, on the rack of a broiler pan coated with cooking spray. Place tarragon sprigs in body cavity of chicken. Arrange potatoes around chicken, and sprinkle 1/4 teaspoon salt and 1/4 teaspoon pepper over potatoes. Insert meat thermometer into meaty part of thigh, making sure not to touch bone.
Bake at 500° for 15 minutes; turn potatoes. Add mushrooms; sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper. Bake at 500° for an additional 15 minutes. Stir mushrooms and bake at 500° for an additional 15 minutes or until thermometer registers 180° and chicken juices run clear. Discard skin.