Heat a dutch oven or large stock pan over medium high heat with olive oil.
Season the chicken with salt and pepper. Add chicken to the pan and brown both sides, about 5 minutes on each side.
While the chicken is browning, peel and chop the apples, onion, and carrots to equal size portions.
When the chicken has browned on each side, remove from the pan and set aside.
Turn down the heat to medium and add the apples, onion, carrots, sage, and thyme. Saute until softened, about 4-5 minutes. Season with salt and pepper.
Add the apple cider vinegar and deglaze until all the liquid has dissolved. Scrape up the brown bits in the pan while liquid is reducing.
Add the chicken back to the pan with the chicken stock and bring to a simmer. Turn the heat to low and cover the pan. Simmer for about 10 minutes or until the chicken is cooked through.
For the dumplings:
While the soup is simmering, make the dumplings.
Heat the milk and butter in a medium saucepan over medium heat.
Mix the flour, parsley, and salt in a separate bowl.
Once the butter has melted and the liquid has just come to a simmer, remove from the heat.
Add the nutmeg, then the dry ingredients to the pan and stir to combine.
Add the eggs one at a time and stir vigorously to combine.
Remove the cover from the dutch oven and take the chicken out and set aside. Season the soup with salt and pepper to taste.
With a spoon, scoop up about one tablespoon of batter at a time and drop it into the soup. Fill the pot with the rest of the dumplings.
While the dumplings are simmering, shred the chicken.
When the dumplings have risen to the surface, they should be done. Taste a dumpling for doneness. They should taste smooth and buttery with no raw flour taste.
Add the chicken back to the pan and serve!