Heat 2 tablespoons of olive oil in a large pot or Dutch oven over medium heat. Add the diced onion, carrots, and celery and sauté for 5–7 minutes until softened, then stir in the minced garlic and cook for 1 more minute.
Add the chicken breasts, chicken broth, thyme, parsley, bay leaf, salt, and pepper to the pot. Bring to a boil, then reduce heat to low, cover, and simmer for 20–25 minutes until the chicken reaches an internal temperature of 165°F (74°C).
Remove the chicken breasts and let rest for 2–3 minutes, then shred using two forks. Return the shredded chicken to the pot and stir in the heavy cream (if using), letting it heat through for 2–3 minutes.
In a mixing bowl, whisk together the flour, baking powder, and salt, then stir in the milk and melted butter until a soft, shaggy dough just comes together — do not overmix.
Drop tablespoon-sized portions of dough into the gently simmering soup, immediately cover with the lid, and cook for 15–20 minutes without lifting the lid until dumplings are cooked through.
Remove and discard the bay leaf, taste and adjust seasoning, then ladle into bowls ensuring each serving has chicken, vegetables, and dumplings. Garnish with fresh chopped parsley if desired and serve immediately.