Heat oil and 2 tbsp butter in a large pot over medium heat.
Add onion, celery, and carrots; cook until softened, about 6 minutes.
Stir in garlic and cook 30 seconds until fragrant.
Sprinkle in 1/4 cup flour; stir for 1 minute to form a paste.
Whisk in broth gradually. Bring to a gentle simmer.
Add chicken, thyme, 1 tsp salt, and pepper. Simmer until chicken is tender, 15 minutes.
Lift out chicken, shred or chop, then return it to the pot.
Make dumplings: whisk 2 cups flour, baking powder, and 1/2 tsp salt in a bowl.
Stir in 3/4 cup milk, melted butter, and parsley to form a thick batter.
Drop rounded tablespoons of batter over the simmering stew.
Cover tightly and cook without peeking until puffed, 12 to 15 minutes.
Stir in the remaining 3/4 cup milk. Simmer 2 minutes and adjust seasoning.