Melt butter in a large Dutch oven over medium heat. Add onion, celery, and carrots. Cook 6 to 8 minutes until onion is translucent.
Add garlic and cook 1 minute, stirring constantly, until fragrant.
Sprinkle flour over vegetables. Stir 1 to 2 minutes until the raw flour smell is gone.
Add chicken thighs, broth, water, bay leaves, and thyme. Bring to a gentle boil, then reduce to a low simmer.
Cover and simmer 20 to 25 minutes until chicken reaches 75°C (165°F) internally and shreds easily when pressed.
Remove chicken and shred with two forks. Discard bay leaves. Return chicken to pot, add heavy cream, season with salt and pepper, and return to a gentle simmer.
In a bowl, whisk flour, baking powder, and salt. Stir in melted butter and milk until just combined. Do not overmix.
Drop batter by rounded tablespoons onto the simmering broth surface, spacing evenly.
Cover tightly. Cook 13 to 15 minutes over medium-low heat without lifting the lid, until dumplings have doubled and a toothpick inserted in the center comes out clean.
Ladle into bowls (2 to 3 dumplings per serving) and garnish with parsley. Serve immediately.
