Melt 3 tablespoons butter in a large Dutch oven or heavy pot over medium; cook onion, celery, and carrots 6-8 minutes until softened.
Sprinkle ⅓ cup flour over vegetables; stir well and cook 2-3 minutes to form a roux.
Gradually whisk in broth constantly to avoid lumps; add thyme, poultry seasoning, salt, and pepper.
Boil; reduce heat to medium-low and simmer 10 minutes.
Meanwhile, whisk 2 cups flour, baking powder, and 1 teaspoon salt in a medium bowl; stir in milk and melted butter just until combined without overmixing.
Drop rounded tablespoons of batter into simmering soup; cover and cook 15 minutes without lifting lid.
Gently stir in chicken and cream; heat through 3-5 minutes.
Adjust seasoning to taste; serve hot with extra black pepper if desired.