Roughly chop onion and carrots. Combine in a large pot with water, peppercorns, bay leaves, parsley and salt. Bring to a gentle simmer and cook for 20 minutes. Add chicken (whole is fine), and simmer for another 20-30 minutes, or until just cooked.
Remove from heat, let the chicken cool in the cooking liquid. Once cool enough to handle, remove the chicken meat from the bones, and shred or dice into small pieces. Place in a large bowl. Reserve the bones for another use (freezing to make more stock in the future is a great idea!). Strain cooking liquid and set aside for walnut sauce.
To Make Walnut Sauce
Process walnuts in food processor until finely chopped. Soak bread in a small amount of chicken cooking liquid. Add to food processor, along with garlic, paprika, coriander, and salt. Process until incorporated. Add enough cooking liquid (chicken stock) until the sauce becomes a pour-able consistency. Taste for salt, and adjust if needed.
To Make Paprika Oil
Warm oil and paprika together in a small pot over medium flame.
To Assemble Dish
Combine walnut sauce with the chicken pieces in a large bowl, and stir to combine. Place in a serving dish, and pour paprika oil over the top. Garnish with walnut pieces, if desired.
Circassian chicken is delicious served on a slice of bread, or alongside freshly chopped veggies such as cucumber or red bell pepper.