Heat oil in a Dutch oven over medium-high heat. Add onion, celery, and carrot; sauté 10 minutes or until tender. Add chicken, broth, water, and thyme; bring to a boil. Reduce heat, and simmer, uncovered, 15 minutes.
While soup simmers, lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, baking powder, rosemary, and salt in a bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Add buttermilk, stirring just until moist.
Drop dough by heaping teaspoonfuls into simmering chicken mixture. Cover and cook over low heat 11 minutes (do not let soup boil).