Heat stock to lukewarm. Combine leeks, celery, parsley, carrots, tarragon, thyme, salt, peppercorns, egg whites, and eggshells and stir into stock. Turn heat to high and stir constantly until mixture comes to a boil (it will become cloudy). As soon as the mixture boils, turn down heat to very low.

A crust will form on top. Make a hole in crust with a ladle to allow liquid to boil through. Ladle liquid over surface of crust several times during first few minutes. Let stock simmer for 30 minutes, then turn off heat. Let stand for 10 minutes.

Line a sieve with damp cheesecloth or a damp, clean kitchen towel. Carefully ladle consomme through sieve, discarding solids. Adjust seasonings. Set consomme aside or let cool and refrigerate until ready to use.

A few minutes before serving, bring consomme to a simmer and add zucchini, squash, and Herb Dumplings. Simmer until dumplings are heated through, 3 to 4 minutes. Garnish with chives, and serve.