Lay a large piece of cheesecloth onto a work surface and put the chicken breast-side down onto it. Gather up the edges and tie them with kitchen string. Put the chicken into the pot, breast-side up and pour over the chicken stock. Bring to a boil. Skim off the foam that rises until no more forms, about 30 minutes. Add the garlic, leeks, bay leaf, tarragon, thyme and some salt and pepper. Turn the chicken breast-side down. Reduce the heat, cover and simmer very gently about 20 minutes more.
Add the pearl onions, carrots, celery, parsnips and turnip. Turn the chicken breast-side up, and continue cooking until the juices in the chicken run clear and the vegetables are tender, about 20 minutes longer. The chicken should pull apart easily. Serve pieces of chicken and vegetables with some broth pooled around.