Place a Dutch oven over medium-high heat. Melt 3 tablespoons of the butter.
Add the onion and celery to the pot and cook for about 4 minutes, stirring occasionally, until softened and fragrant.
Sprinkle ¼ cup of the flour over the vegetables. Stir continuously for about 1 minute to cook out the raw flour flavor.
Slowly pour in the chicken broth while whisking to create a smooth base. Bring the mixture to a gentle boil.
Reduce the heat and simmer for 5 minutes until the broth begins to thicken slightly.
Stir in the cooked chicken, green beans, and peas and carrots. Season with salt and black pepper. Continue simmering for another 5 minutes.
While the soup cooks, prepare the dumpling dough. In a mixing bowl, combine the remaining 1 cup flour, baking powder, a pinch of salt, and the chopped parsley.
Add the remaining 2 tablespoons cold butter and work it into the flour mixture using a fork or pastry cutter until small crumbs form.
Pour in the milk and stir just until a thick dough comes together. Do not overmix.
Using a small cookie scoop or two spoons, drop portions of dough directly onto the simmering soup.
Cover the pot tightly and cook for 12 to 15 minutes without lifting the lid. The trapped steam helps the dumplings become light and fluffy.
Check that the dumplings are cooked through, then remove the pot from the heat.
Ladle the soup and dumplings into bowls and sprinkle with fresh parsley before serving.