Dry chicken pieces with a paper towel, season them with salt and pepper, then place in a large, heavy pot over medium-high heat.
Cook chicken on both sides until nicely browned, then transfer to a plate and set aside.
Reduce the heat to medium and add the butter to the pot. Once melted, add the carrots, celery, and onion and cook for about 5 minutes.
Add the Italian seasoning, garlic powder, onion powder, and flour to the pot, and continue cooking until fragrant, about 1 minute.
Return the chicken to the pot and add the chicken broth. Stir, cover, and simmer over medium-low heat for 20 minutes.
Meanwhile, start making the dumplings by whisking the dry ingredients together.
Add the milk to the dry mixture and stir until just combined. (If it's too dry, add another splash of milk.)
After simmering the soup, remove the chicken and use two forks to remove the meat from the bones. Return the shredded chicken to the pot, add the milk, and stir well.
Use a spoon to transfer dumpling-sized portions of dough to the soup, continuing until you've used up all the dough. Cover the pot and simmer for another 15 minutes, then serve warm.