Bring 3 quarts of water to a boil and add 1 tablespoon salt. Set up an ice bath nearby. Blanch the peas in the water for 2 minutes, then remove with a slotted spoon, and immediately refresh in the ice bath.
Drain and set aside.
In a soup pot, bring to chicken stock to a simmer. In a separate pot, combine the blanched peas, breadcrumbs, eggs, cheese, and salt and pepper, to taste, and stir well. Stir in a ladle of the chicken stock and cook over low heat, stirring constantly and gradually adding the warm stock, a ladle at a time, to keep the eggs from coagulating. Adjust the seasoning and serve.
In a large, heavy-bottomed saucepan, heat the oil over high heat until smoking.
Add all the chicken parts and brown all over, stirring to avoid burning.
Remove the chicken and reserve.
Add the carrots, onions, and celery to the pot and cook until soft and browned. Return the chicken to the pot and add 3 quarts of water, the tomato paste, peppercorns, and parsley. Stir with a wooden spoon to dislodge the browned chicken and vegetables bits from the bottom of the pan. Bring almost to a boil, then reduce heat and cook at a low simmer until reduced by half, about 2 hours, occasionally skimming excess fat.
Remove from heat, strain, and press on the solids with the bottom of a ladle to extract out all liquids. Stir the stock to facilitate cooling and set aside. Refrigerate stock in small containers for up to a week or freeze for up to a month.
