Make the stock: Combine all of the ingredients in a large stockpot and bring to a boil over high heat. Reduce the heat to a simmer and cook, skimming occasionally, until slightly reduced, 4 to 5 hours. Strain the solids and discard. Set the stock aside and keep warm until you're ready to add it to the soup. (You could also make the stock in a pressure cooker.)
Make the dumplings: Whisk together the eggs, milk, nutmeg and chives in a medium bowl. Season well with salt and pepper then stir in the flour until smooth. Cover and allow to rest until you're ready to add to the dumplings to the soup.
Make the soup: Place a large stockpot over medium-high heat. Melt the butter in a large stockpot over medium-high heat and add a drizzle of olive oil. Add the onions, carrots, celery and a large pinch of salt and cook, stirring occasionally, until softened, 5 to 8 minutes. Stir in the cabbage and season with salt and pepper. Cook, stirring occasionally, until the cabbage begins to wilt 2 to 3 minutes. Add the turkey stock and bring to a simmer. Simmer until all of the flavors come together, about 45 minutes.
Increase the heat to bring the soup to a gentle boil. Place the dumpling batter on a damp cutting board and cut into bite-size pieces with a bench scraper or knife. Drop the dumplings into the boiling soup and cook until they all to float to the surface. Continue to boil for 1 minute, then stir in the dill and serve.