Place first 4 ingredients on a double layer of cheesecloth. Gather edges of cheesecloth together, and tie securely. Place cheesecloth bag, potatoes, and the next 7 ingredients (potatoes through black pepper) in a large Dutch oven or stockpot over medium-high heat. Bring to a boil; reduce heat, and simmer, uncovered, 25 minutes. Turn chicken over and cook 20 minutes or until vegetables are tender and chicken is done.
Remove chicken and vegetables with a slotted spoon; set aside, and keep warm. Discard the cheesecloth bag. Return the broth mixture to a boil, and cook until reduced to 4 cups (about 15 minutes). Remove from heat. Reserve 3 cups broth mixture for another use.
Melt the butter in a saucepan over medium heat. Stir in the flour; reduce heat, and cook 1 minute. Add 1 cup broth mixture, and stir with a whisk. Stir in milk. Cook until thick (about 8 minutes), stirring frequently.
Remove meat from bones; discard bones and skin. Chop chicken. Arrange vegetables on a platter, and top with chicken. Spoon sauce over chicken and vegetables. Sprinkle with chives, if desired.