In a large pot or Dutch oven, heat the olive oil over medium heat. Add the diced onion, carrots, and celery. Sauté for about 5-7 minutes until the vegetables are softened.
Stir in the minced garlic, thyme, rosemary, salt, and pepper, cooking for an additional minute until fragrant.
Pour in the chicken broth and bring the mixture to a boil. Add the chicken breasts and reduce the heat to a simmer. Cover and cook for about 20 minutes, or until the chicken is cooked through.
While the chicken cooks, prepare the dumpling dough. In a mixing bowl, whisk together the gluten-free flour, baking powder, salt, and pepper.
In another bowl, combine the milk, melted butter, and egg. Add the wet ingredients to the dry ingredients, mixing until just combined. The dough will be thick.
Once the chicken is done, remove it from the pot and shred it with two forks. Return the shredded chicken to the pot and stir in the frozen peas.
Using a spoon, drop dollops of the dumpling dough onto the surface of the simmering broth. Cover the pot and cook for an additional 15 minutes without lifting the lid.
Serve the chicken and dumplings hot, using a ladle to scoop the broth and dumplings into bowls.