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 the kitchen string. Put the chicken into the pot, breast-side up and pour over chicken stock. Bring to a boil. Skim off the foam that rises until no more forms, about half an hour.
Add the bay leaf, thyme, salt, garlic and leeks. Turn the chicken, breast side down. Cover, and simmer very gently about 20 minutes more. Add the remaining vegetables, 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 will be so tender if pulls apart easily. Serve pieces of chicken and vegetables with some broth pooled around.