Start by boiling the chicken breasts in a pot filled with water until fully cooked, roughly 15 minutes. Once done, shred the chicken and set aside.
In a large pot, melt butter over medium heat. Add diced onions, carrots, and celery. Stir occasionally and cook until onions become translucent.
Pour chicken stock into the pot. Increase the heat and bring it to a boil.
Once boiling, add the shredded chicken, egg noodles, salt, and pepper. Stir well.
For the dumplings, in a mixing bowl, combine flour, baking powder, salt, and milk. Mix until a thick dough forms.
Using a spoon, drop dollops of dumpling dough into the boiling soup. Continue cooking for another 15 minutes.
Reduce the heat to low, and let the chicken noodle and dumpling soup simmer for another 15 minutes, allowing all flavors to meld beautifully.
Taste and adjust the seasoning if necessary. Once satisfied, turn off the heat.