Combine 1 cup potato, turnips, and 1 can of broth in a small saucepan, and bring to a boil. Reduce heat; simmer 8 minutes or until tender. Combine potato mixture and milk in a blender or food processor; process until smooth.
Heat the butter in a large, heavy saucepan over medium-low heat. Add leek, thyme, and garlic; sauté 12 minutes or until leek is tender. Add 1 cup of potato, 1 can of broth, and carrot. Cover and simmer 8 minutes. Stir in the greens and chicken. Cover and simmer 12 minutes or until chicken is done. Stir in the potato purée, vinegar, pepper sauce, salt, and pepper; simmer, uncovered, 6 minutes.