Wash & salt two large chicken breasts. Preheat large cast iron stock pot over medium-high heat with 1T olive oil in the bottom. Add chicken when the pot is hot (or when olive oil is shimmering). Cook on one side for 5-7 minutes until brown and chicken lifts easily. Flip to cook second side for another 5-7 minutes or until center is no longer pink. Remove from pan. (You can also use a rotisserie chicken as a shortcut.)
While chicken is cooking, peel and dice carrots and onion. Wash and chop celery. Add carrots, celery, onion, celery, garlic, and 4T of butter once chicken is removed from pot. Stir and saute until vegetables are soft.
Add in 1/2 cup flour and stir until veggies are coated in flour. (For about 1 minute). Add chicken stock and wine. Stir and scrape bottom of pot with spoon to lift any stuck-on bits. Bring to a boil and allow to thicken.
While soup is thickening, make dumpling dough. Put 2 cups of flour, 2t salt, and 2t baking powder into a clean mixing bowl. Melt 4T of butter and add to dry ingredients. Stir until dough is crumbly. Add 1 cup of whole milk. Stir until just combined. (It'll look lumpy.) Set aside.
Once cooled, shred chicken with a fork or your fingers.
Once soup has thickened, reduce heat to a simmer. Add chicken and 1 cup of whole milk. Stir.
Drop dumpling dough into soup 1T at a time. A cookie dropper is perfect for this, but you can use two spoons as well. Each dumpling will sink and then rise. Keep adding dumplings until all the dough is used. Cover and cook for a remaining 15 minutes.
Salt & Pepper to taste. Top with fresh parsley to serve.