For the stock:
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 scum as it cooks.
When done remove the chicken to a cutting board. Strain the stock and shred the meat into big pieces - the stock will be used for the sauce and the chicken will be folded into it.
For the dumplings:
Sift the dry ingredients together in a large bowl. In a small bowl, using a whisk, lightly beat the eggs, chives and buttermilk together; pour the liquid ingredients into the dry ingredients and gently fold. Mix just until the dough comes together; the batter should be thick and cake-like.
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 to coat the flour and remove the starchy taste. 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 spoon, about 15 minutes. Stir in heavy cream.
Fold the reserved shredded chicken into the sauce and bring up to a simmer. 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 touching or crowded. Let the dumplings poach for 10 to 15 minutes until they are firm and puffy. Remove and discard the bay leaves. Season with freshly cracked black pepper and garnish with chopped chives before serving.