Heat butter and olive oil in a large pot over medium heat.
Add diced onion, sliced carrots, and chopped celery. Cook for 5 minutes until softened.
Add minced garlic and cook for 30 seconds until fragrant.
Sprinkle in 3 tablespoons of flour and stir for 1 minute to thicken.
Gradually whisk in chicken broth and milk to create a smooth base.
Add shredded chicken, thyme, black pepper, and salt to taste. Simmer for 10 minutes.
In a separate bowl, combine flour, baking powder, and salt for dumplings.
Cut in cold diced butter, then stir in milk until dough forms.
Drop spoonfuls of dumpling dough onto the simmering soup.
Cover the pot and cook undisturbed for 12 to 15 minutes until dumplings are fluffy.
Sprinkle chopped parsley over the soup and serve hot.