In a small saucepan, gently heat the walnut oil with 4 teaspoons of the paprika. Remove from the heat and let infuse.
In a medium pot, place the chicken, carrots, celery, half of the onion, the garlic clove, bay leaves, parsley stems and peppercorns. Cover with water by 1 to 2 inches (about 1 quart) and bring to a boil. Reduce the heat and simmer until the chicken is tender, about 20 minutes. Remove from the heat and remove the chicken to a plate to cool. Strain the stock into a clean container and let cool.
When the meat is cool enough to handle, shred or chop into pieces.
In a small bowl, soak the bread with enough stock to soften. Let sit for 2 minutes then squeeze dry and tear into pieces.
Chop the remaining onion half.
In a large skillet, heat the olive oil over medium heat. Add the chopped onions, garlic, remaining 2 tablespoons paprika, and the cayenne and gently cook, stirring, until aromatic and soft but not colored, 5 to 6 minutes. Remove from the heat and let cool.
In a food processor, grind the walnuts. Add the onion mixture, bread pieces and enough of the reserved stock to blend, about 1 cup. Process to make a smooth paste, adding additional stock 1 tablespoon at a time as needed. Adjust the seasoning, to taste, with salt and pepper.
In a bowl, toss the chicken pieces with half of the sauce to coat. Place on a decorative platter and spread the rest of the sauce over the top.
Drizzle with the paprika-infused walnut oil and garnish with parsley. Serve at room temperature with bread.
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