Melt 2 tablespoons butter in a slow cooker set to sauté or a sauté pan over medium-high heat. Add carrots, celery, and onion, and sauté 5 minutes.
If using a pan, transfer vegetables to slow cooker. Lay chicken thighs on top. Mix together thyme, salt, rosemary, and pepper, and sprinkle evenly over the chicken. Pour in the chicken broth and add the bay leaves.
Cover and cook on Low for 6 hours.
Just before the time is up, make the dumplings. In a bowl, mix together the flour, parsley, baking powder, and salt. Mix in the milk, egg, and melted butter until a sticky, thick batter forms.
Take the lid off the slow cooker and use tongs or two forks to shred the chicken.
Scoop batter and place on top of the chicken. Leave room between each dumpling to allow for expansion.
Replace the lid and turn the slow cooker to High. Cook for 1 hour, until dumplings have risen and are cooked through.
Notes
This can be cooked on high for less time if desired.
Make sure to reserve 1 hour of time for cooking the dumplings.