Heat olive oil in a large pot over medium heat. Add the diced onion, carrots, and celery. Sauté until vegetables soften, about 5 minutes. Stir in the garlic and cook for an additional 1 minute.
Pour in the chicken broth, and add the chicken breasts, thyme, rosemary, salt, and pepper. Bring to a boil, then reduce heat to low. Cover and simmer until the chicken is cooked through, about 20 minutes.
Remove the chicken from the pot and shred it using two forks. Return the shredded chicken to the pot and add the frozen peas. Stir to combine and maintain a low simmer.
In a mixing bowl, whisk together the flour, baking powder, and salt. Gradually add in the milk and melted butter, mixing until just combined.
Drop spoonfuls of the dumpling batter over the chicken mixture in the pot. Cover and cook for an additional 15 minutes without lifting the lid, until the dumplings are cooked through and puffed up.
Taste and adjust seasoning if needed. Serve hot, and enjoy!
For a richer flavor, consider adding a splash of cream to the broth before serving.