1. For the stock: Combine all stock ingredients in a large stock pot and bring to a boil, then reduce heat and simmer skimming off any impurities that rise to the surface. Simmer for 3 hours. Strain through a fine mesh strainer and discard solids.
2. For the Dumplings: add flour to a large bowl. Make a well in the flour. In a separate bowl, whisk together eggs and milk. Add the eggs and milk mixture to the well in the flour, then add the chives and whisk all together until smooth.
3. For the soup: Heat butter over medium heat in a large sock pot. Add the vegetables, and sweat the vegetables until they are translucent. Add the turkey meat and stock to the pot; cover and bring to a simmer. Continue to simmer for 45 minutes.
4. Bring soup to a boil, scoop dumplings into boiling broth with smallest kitchen scoop. Wait for them to float to the surface. Let boil another minute. Add tarragon, and serve.