Cover potatoes with cold salted water by 1 inch and simmer until tender, 20 to 25 minutes.
Drain and keep warm.
While potatoes are boiling, sauté carrots, celery, and onion in oil in a 3-quart heavy saucepan over high heat, stirring frequently, until beginning to brown.
Add broth and bring to a boil. Simmer, covered, until vegetables are tender, about 13 minutes. Stir in beets and their brine and simmer, covered, 8 minutes more. Ladle borscht into bowls and add potatoes. Top with sour cream and dill.
