Stock: combine the stock ingredients in a large stockpot, bring to a boil, then simmer 5 hours, skimming impurities. Strain. You need 1 1/2 gallons.
Dumplings: make a well in the flour. Whisk the eggs and milk, pour into the well with the nutmeg and chives and whisk until smooth.
Soup: heat the butter in a large stockpot over medium heat and sweat the vegetables with a large pinch of salt until translucent. Add the turkey and 1 1/2 gallons stock, cover, bring to a simmer and simmer 45 minutes.
Bring to a boil. Spread the batter on a damp cutting board and cut off bite-size dumplings with a bench scraper or knife into the boiling soup. When they all float, boil 1 minute more, add the tarragon and serve.