Melt the butter in a 6-quart pot over medium heat. Add the garlic, carrots, celery root, turnip, rutabaga, onion, diced fennel and a large pinch of salt; cook until the vegetables are tender, about 5 minutes.
Add the wine, 6 cups water, fennel seeds, thyme and bay leaves; bring to a simmer. Add the chicken and simmer for 2 hours. Remove the chicken and let cool. Pick off the meat in shreds; set aside.
About 15 minutes before the chicken has finished simmering, prepare the dumplings: Combine the milk, chicken fat, 1 tablespoon tarragon, the nutmeg and 1/2 teaspoon each salt and pepper in a medium saucepan; bring to a simmer over medium heat. Remove from heat; add the flour and stir vigorously with a wooden spoon until the liquid is absorbed. Whisk in the eggs, one at a time, to make a thick dough.
Bring the soup to a boil. Form the dough into eight dumplings (an ice cream scoop is perfect for this); drop into the soup. Cover and cook until the dumplings float to the surface, about 10 minutes.
Add the chicken meat, parsley and the remaining 1 tablespoon tarragon to the soup; stir carefully and divide among warm bowls.