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; saut 6 minutes or until onion is tender.
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).
