Warm the oil in a large pot over medium heat. Soften onion, carrots, and celery for 6–8 minutes.
Stir in garlic and thyme. Cook until fragrant, about 30 seconds.
Add chicken, salt, and pepper. Cook, stirring, until the edges turn opaque, 3–4 minutes.
Pour in the broth and bring to a gentle simmer. Cook until the chicken is tender, about 15 minutes.
Whisk flour and water in a small bowl. Stir the slurry into the pot and simmer 3–4 minutes to thicken.
Stir in the milk and peas. Taste and adjust seasoning. Keep at a steady simmer.
Make the dumpling batter: whisk flour, baking powder, and salt. Add milk and melted butter; stir just to combine. Fold in parsley.
Drop heaping tablespoon scoops of batter over the simmering stew. Cover tightly and cook 12–14 minutes without lifting the lid.
Uncover and check one dumpling; it should be cooked through. Rest 5 minutes off the heat, then serve.