Place chicken in soup pot and cover with cold water. Add the carrots, celery, onion, garlic, bay leaves, and peppercorns. Bring to a boil, and then reduce heat to a simmer for 1 to 1 1/2 hours, skimming impurities from the top throughout.
Remove the chicken from the stock, split in half and allow let cool on a sheet pan for 15 minutes.
Remove meat from carcass and set aside. Return the bones to the simmering pot of stock and cook for one hour. Tear the meat into medium-large pieces and reserve.
While the stock is cooking, prepare the dumpling dough. In medium bowl, combine flour and salt. Add eggs and mix until a soft dough forms. Wrap dough tightly in plastic wrap and place in the refrigerator for at least an hour.
Strain stock through a fine mesh strainer, return to pot and keep at a simmer. Discard solids.
Remove dough from the refrigerator. Dust work surface and rolling pin with flour. Roll dough out to about 1/8 inch thickness. Using a pizza cutter, cut the dough into 1-inch-by-2-inch strips of dough. Individually drop dough strips into simmering stock and cook for about 10 minutes.
In separate large pot over medium-low heat, melt the butter and add remaining carrots, celery, and onions. Sweat vegetables for 4 to 5 minutes. Stirring constantly, sprinkle flour over the sweating vegetables and mix well.
Add dumplings to the stock, stirring until well incorporated.
Return to simmer, stirring occasionally, for about 10 minutes or until mixture thickens slightly. Season with salt and pepper.
Stir in reserved chicken and peas and simmer until heated through.
Serve hot and enjoy!
Chef's Tip: Fresh biscuits go nicely with this dish.