Put broth, chicken, and 2 cups water in a large pot. Bring to a boil, then simmer 1 hour, skimming fat from broth as it cooks.
Mix eggs in a large bowl with a fork just to break up a little. Add oil, club soda, matzo meal, pepper, and 1/2 tsp. salt and mix well. Chill, covered, at least 30 minutes and up to 2 hours.
Remove chicken from broth (let broth continue to simmer). Let chicken cool about 15 minutes, then tear meat into shreds and discard bones.
Shape matzo dough into 1-in. balls using wet hands; add each ball to broth as you go. Cover broth and matzo balls and simmer 30 minutes. Add carrots, onion, celery, and remaining 1/2 tsp. salt and cook 5 minutes. Stir in chicken and heat until warm, 1 minute. Ladle soup into bowls and garnish with dill.
Note: Nutritional analysis is per 1 1/2-cup serving.