Tie half of the parsley together with twine. Place in a large pot over medium heat along with the broth, cinnamon stick, peppercorns, allspice, coriander and 1 1/2 teaspoons salt. Cover and simmer 10 minutes.
Tie the leeks together with twine and add to the pot. Add the carrots, turnips and water to cover, if necessary. Bring to a boil, then reduce to a simmer and cook until the vegetables are tender, about 15 minutes. Transfer the vegetables to a bowl with a slotted spoon; untie the leeks.
Add the chicken to the pot, cover and poach over low heat until firm, about 20 minutes. Transfer the chicken to a cutting board, reserving the broth; remove the skin and shred the meat.
Heat the olive oil in a skillet over medium heat. Add the winter greens and cook until wilted, about 5 minutes. Add 1 cup of the reserved broth and cook until the greens are tender, about 5 more minutes. Season with salt.
Strain the remaining broth, return to the pot and simmer 10 minutes. Meanwhile, chop the remaining half of the parsley. Season the broth with salt and pepper. Add the leeks, carrots and turnips and heat through, then divide among bowls along with the chicken and greens. Ladle the broth on top. Sprinkle with the chopped parsley.