In a medium saucepan, combine broth and chicken; bring broth to a gentle simmer over medium heat. Reduce heat slightly so liquid is moving but not simmering; cook until chicken is cooked through, about 15 minutes. Transfer chicken to a plate and let rest until cool enough to handle. Shred into bite-sized pieces.
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Bring broth to a rapid simmer; add rice and cook just until tender, 16 to 18 minutes. Return chicken to pot. In a medium bowl, whisk eggs until frothy; whisk in lemon juice. Whisking constantly, slowly pour 1 cup hot broth into egg mixture. Immediately whisk egg mixture into pot. Season soup with salt and pepper. Serve immediately, garnished with pepper and dill.
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