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.