Warm the olive oil in a large heavy pot over medium heat.
Sauté onion, carrots, and celery until softened, 5 to 6 minutes.
Stir in garlic and thyme; cook until fragrant, about 30 seconds.
Sprinkle flour over the vegetables; stir for 1 minute to form a light roux.
Whisk in the broth gradually. Add chicken and bring to a gentle boil.
Reduce to a simmer, cover, and cook until chicken is cooked through, 15 to 18 minutes.
Transfer chicken to a board; shred with two forks. Return meat to the pot.
Stir in peas, milk, salt, and pepper. Keep the soup at a steady simmer.
Make dumpling batter: whisk flour, baking powder, salt, and parsley in a bowl.
Add milk and melted butter; stir just until a thick, sticky dough forms.
Drop heaping tablespoons of dough onto the simmering soup. Cover and cook 12 to 14 minutes without lifting the lid.
Uncover, taste, and adjust seasoning. Let rest 5 minutes before serving.