We made our matzo balls small, for easier eating, but if you prefer 1 large ball per serving, divide the dough into 4 to 6 balls and boil them 10 minutes longer.
1. 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.
2. 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.
3. Remove chicken from broth (let broth continue to simmer). Let chicken cool about 15 minutes, then tear meat into shreds and discard bones.
4. 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.
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