To make the soup, place oil in a heavy bottom soup pot over medium-high heat.
Season chicken thighs with salt and pepper on both sides. Place in the soup pan to brown, 2 to 3 minutes on each side. Remove from the pan and place on a plate. Chicken won’t be fully cooked through but it will return to the pot to simmer a bit later.
Add the onion, carrot, celery, garlic, and thyme to the pot. Stir and saute until the onions are cooked through and translucent, about 5 minutes.
Return the chicken to the pot. Add the bay leaf and chicken stock.
Reduce heat to low and simmer, uncovered, until the vegetables are cooked through and softened, about 30 to 40 minutes.
Remove the chicken when it is cooked through and carefully chop and shred it. Return the chicken to the pan.
While the soup simmers. Make the dumplings. In a medium bowl, whisk together flour, baking powder, salt, thyme, and pepper. Add the butter and use your fingers to break it down into the flour mixture creating small flecks of butter throughout the mixture.
Add the beaten egg and buttermilk and stir until just combined. Try not to over-stir the dumpling dough.
When the vegetables in the soup are cooked through and the chicken is chopped and returned to the pot, bring the soup to a light simmer and season to taste. Stir in peas and parsley. Use a small spoon to gently dollop about 2 tablespoons of dough into the simmering soup.
The dumpling may sink, but will rise to the surface of the soup as it cooks. Add all of the dumplings.
Cover the pot. Allow to simmer for 10 minutes.
After 10 minutes the dumplings should be cooked through. Remove from heat and serve warm!