Place the chicken and all stock ingredients in a large Dutch oven and cover with water. Set over medium-high heat and bring to a boil. Reduce heat to a simmer and cook for 1 hour until the chicken is tender. Skim the surface of fat and foam as it cooks.
When done remove the chicken to a cutting board. Strain and reserve the stock and shred the meat into pieces.
Make the dumplings: In a large bowl, sift the dry ingredients together.
In a small bowl, using a whisk, lightly beat the eggs, chives, parsley and buttermilk together; pour the liquid ingredients into the dry ingredients and gently fold. Mix just until the dough comes together.
To prepare sauce: In a Dutch oven, over medium heat, add the butter and oil. Add the carrot, celery, garlic, and bay leaves and saute until the vegetables are soft, about 5 minutes.
Stir in the flour to make a roux. Continue to stir and cook for 2 minutes.
Slowly pour in the chicken stock, 1 cup at a time, stirring well after each addition.
Add frozen peas and pearl onions.
Let sauce simmer until it is thick enough to coat the back of a wooden spoon, about 15 minutes.
Fold the reserved shredded chicken into the sauce and bring up to a simmer.
Remove and discard the bay leaves.
Using 2 spoons carefully drop heaping tablespoonfuls of the dumpling batter into the hot mixture. The dumplings should cover the top of the sauce, but should not be too crowded.
Let the dumplings poach for about 10 minutes until they are firm and puffy Turn the dumplings over and let poach another 5 minutes.
Season with freshly cracked black pepper.
Garnish with chopped chives and parsley before serving.