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Combine chicken, carrots, peas, and celery in a saucepan; add water to cover and bring to a boil. Boil for 15 minutes, then remove from the heat and drain.
While the chicken is cooking, melt butter in another saucepan over medium heat. Add onion and cook until soft and translucent, 5 to 7 minutes. Stir in flour, salt, pepper, and celery seed
Slowly stir in chicken broth and milk.
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