FOR CHICKEN AND STOCK:
PLACE chicken, onion, carrot, celery, thyme, bay leaves, and peppercorns in large stock pot. Cover chicken with water. Bring to a boil on medium high heat. Reduce heat and simmer until chicken is tender. Remove chicken from pot and set aside to cool. Strain stock into another stockpot or large bowls; set aside to cool. Remove layer of fat. Reserve chicken stock. Shred chicken (removing all meat from the bones); set aside.
FOR HOT PEPPER AND WALNUT SAUCE:
MELT butter in large skillet over medium heat; add onions and cook until soft and translucent. Meanwhile, in medium bowl, soak the bread in evaporated milk. Add the garlic to the skillet and cook for another minute. Stir in palillo and pepper paste until well blended. Add the soaked bread with milk into the skillet, stirring constantly for 30 seconds. Add the 1 cup of stock*, walnuts, Parmesan cheese and salt, stirring occasionally until blended and sauce thickens (about 10 minutes).
ADD shredded meat to sauce; let simmer for 10 more minutes. Serve with rice or boiled new or Yukon gold potatoes. Garnish with sliced eggs and olives.
* Additional stock can be added if a thinner sauce is desired. Freeze remaining stock for another use.