In a large pot or Dutch oven, heat olive oil over medium heat. Add onion, carrots, and celery; cook, stirring occasionally, until softened, about 5 minutes. Add garlic, thyme, and rosemary; cook 1 minute more.
Pour in chicken broth and add chicken breasts. Bring to a boil, then reduce heat to low, cover, and simmer until chicken is cooked through, about 15 minutes. Transfer chicken to a plate and shred with two forks.
While chicken cooks, make dumpling batter: In a medium bowl, whisk together flour, baking powder, salt, and pepper. Stir in milk and melted butter until just combined (do not overmix).
Return shredded chicken to the pot. Stir in frozen peas and parsley. Bring soup to a gentle simmer.
Drop tablespoonfuls of dumpling batter onto the surface of the simmering soup. Cover and cook without lifting the lid for 15 minutes, until dumplings are puffed and cooked through.
Serve hot, garnished with additional parsley if desired.
Tips
For a richer broth, use chicken thighs instead of breasts.
Do not overmix the dumpling batter to keep them light and fluffy.
Leftovers can be stored in an airtight container in the refrigerator for up to 3 days; dumplings may soften upon reheating.