Combine first 5 ingredients in a large Dutch oven over medium-high heat; bring to a boil. Reduce heat, and simmer, uncovered, 15 minutes or until chicken is done. Remove pan from heat. Remove chicken pieces from broth; cool slightly. Strain broth through a sieve into a large bowl; discard solids. Remove chicken from bones. Discard bones; chop chicken into bite-sized pieces. Set chicken aside.
Heat oil in pan over medium-high heat. Add carrots, celery, onion, and garlic; sauté 6 minutes or until onion is tender. Add reserved broth mixture and 1/2 teaspoon salt; simmer 10 minutes. Discard thyme sprigs. Keep warm.
To prepare dumplings, lightly spoon flour into dry measuring cups; level with a knife. Combine 1 1/4 cups flour, rosemary, baking powder, and 1/4 teaspoon salt in a large bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Combine buttermilk and egg, stirring with a whisk. Add buttermilk mixture to flour mixture, stirring just until combined.
Return chopped chicken to the broth mixture; bring to a simmer over medium-high heat. Combine 1/4 cup flour and 1/4 cup water, stirring with a whisk until well blended to form a slurry. Add flour mixture to pan; simmer 3 minutes. Drop the dumpling dough, 1 tablespoon per dumpling, onto chicken mixture to form 12 dumplings. Cover and cook 7 minutes (do not let broth boil). Sprinkle with black pepper.