Heat butter in a soup pot over medium heat.Add onions, celery, and mushrooms; toss to coat. Add turnips, carrots, and radishes; stir to combine. Add chicken, stock, and 4 cups water, or enough to cover. Bring to a boil; reduce to a simmer. Cook until vegetables are just tender and chicken is cooked through, about 15 minutes.
Fill a large bowl with water and ice; set aside.Transfer vegetables, chicken, and the stock to a bowl set in the ice bath; cool to room temperature. Cover; chill overnight.
When ready to serve, wash romaine, cut into bite-size pieces, and arrange on a platter. In a bowl, whisk vinegar,mustard, and honey. Slowly whisk in oil; set aside.
Discard any fat that has accumulated on top of chicken mixture. Shred chicken; place in a bowl. Drizzle a third of the vinaigrette over chicken. Sprinkle with 1/2 teaspoon salt and 1/8 teaspoon pepper; toss.
Arrange chicken over greens. Remove vegetables from stock; cut into bite-size pieces. Place in bowl used to toss chicken. Drizzle with half the remaining vinaigrette, sprinkle with remaining 1/2 teaspoon salt and 1/8 teaspoon pepper, toss, and arrange vegetables around chicken on platter. Drizzle with remaining vinaigrette; serve.