In a large pot or Dutch oven, heat a tablespoon of oil over medium heat. Add the diced onion and sauté until translucent, about 3-4 minutes.
Stir in the minced garlic, carrots, and celery. Cook for another 5 minutes, until the vegetables are slightly softened.
Add the chicken to the pot, followed by the chicken broth, thyme, parsley, salt, and pepper. Bring the mixture to a boil, then reduce the heat to low and let it simmer for 20 minutes, or until the chicken is fully cooked.
Remove the chicken from the pot and shred it using two forks. Return the shredded chicken to the pot along with the frozen peas.
While the soup is simmering, prepare the dumplings. In a medium bowl, combine flour, baking powder, and salt. Stir in the milk and melted butter until just combined (do not overmix).
Drop spoonfuls of the dumpling batter on top of the simmering soup. Cover the pot and let the dumplings cook for 10-12 minutes, or until they are fluffy and cooked through.
Once the dumplings are ready, serve the soup hot, garnished with fresh parsley.