Heat olive oil in a large Dutch oven or heavy-bottomed pot over medium heat. Add onions, carrots, and celery, sautéing until the onions become translucent (about 5 minutes). Add garlic and cook for 30 seconds more until fragrant.
Sprinkle in the thyme, rosemary, salt, and pepper, stirring to coat the vegetables evenly. Add the flour and stir constantly for 2 minutes to cook out the raw flour taste.
Gradually pour in the chicken broth while stirring continuously to prevent lumps from forming. Add bay leaves and bring the mixture to a gentle boil. Reduce heat to maintain a simmer.
Add chicken pieces to the simmering broth, ensuring they're fully submerged. Cover and cook for about 15-20 minutes until the chicken reaches an internal temperature of 165°F (74°C). Once cooked, remove the chicken and shred it using two forks.
While the chicken cooks, prepare your dumplings. In a medium bowl, whisk together flour, baking powder, and salt. In a separate small bowl, combine milk and melted butter. Pour the wet ingredients into the dry ingredients and stir until just combined.
Return the shredded chicken to the pot and stir in the heavy cream or milk. Taste and adjust seasoning if needed.
Using a tablespoon or cookie scoop, drop dumpling dough directly onto the simmering broth, leaving some space between them as they'll expand during cooking.
Cover the pot and reduce heat to a low simmer. Cook undisturbed for 15 minutes, until the dumplings are cooked through. They should be light, fluffy, and nearly doubled in size. A toothpick inserted into the center of a dumpling should come out clean.
Remove bay leaves before serving. Garnish with fresh parsley if desired.