In a blender or food processor, combine 3 cups cooked chicken, eggs, 3/4 cup chicken broth, 1 cup flour, parsley, 2 teaspoons salt, pepper and tarragon.Process until smooth.
In a large pot, bring 4 cups chicken broth to a boil.Drop dumpling mixture by rounded spoonfuls into boiling broth.Simmer, uncovered, 5 to 8 minutes, until well formed and slightly browned.Remove with a slotted spoon and drain on paper towels.Reserve dumpling cooking liquid.
In a large pot melt butter over medium heat.Cook onion, celery and garlic in butter until onion is translucent.Stir in 1/2 cup flour until fully incorporated.Pour in 2 quarts chicken broth, stirring constantly.Bring to a boil.Introduce 1 1/2 teaspoons salt and carrots.Cover, reduce heat, and simmer 15 minutes.
Stir in dumplings, reserved liquid and 3 cups cooked chicken.Simmer 15 minutes more before serving.
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