To prepare broth, combine first 7 ingredients in a large stockpot; bring to a boil. Reduce heat to medium-low, and simmer 2 hours, skimming as necessary. Remove chicken from broth; cool. Remove meat from bones. Shred meat; set aside. Discard bones. Strain broth through a sieve over a bowl; discard solids. Place broth in a large saucepan; bring to a boil. Cook until reduced to 6 cups (about 8 minutes).
To prepare dumplings, weigh or lightly spoon 5 ounces (about 1 cup) flour into a dry measuring cup; level with a knife. Combine 4 ounces (3/4 cup) flour, baking powder, and 1/4 teaspoon salt; stir with a whisk. Cut in butter with a pastry blender or two knives until mixture resembles coarse meal. Add buttermilk; stir to combine. Turn dough out onto a lightly floured surface; knead 5 times, adding remaining 1 ounces (1/4 cup) flour as needed. Divide mixture into 24 equal portions.
Heat a large Dutch oven over medium-high heat. Coat pan with cooking spray. Add chopped onion, 1 cup celery, and 3/4 cup carrot to pan; sauté 4 minutes, stirring occasionally. Add broth and 3/4 teaspoon salt; bring to a boil. Reduce heat, and simmer 20 minutes or until vegetables are tender. Drop dumplings into pan; cover and cook 10 minutes or until dumplings are done, stirring occasionally. Remove 1/4 cup liquid from pan; stir in 1 tablespoon flour. Return chicken to pan. Add flour mixture to pan; bring to a boil. Cook 1 minute or until slightly thick, stirring occasionally. Remove from heat; stir in chives.