For the picada: Toast the nuts in a small skillet, then remove to a plate. Add the EVOO to the skillet and toast the bread slices until golden. Chop the bread and add to a food processor along with the parsley, lemon juice, garlic, toasted almonds and a splash of water. Pulse until the mixture is a pesto-like consistency. Refrigerate until ready to serve.
For the stew: Heat the EVOO in a Dutch oven over medium-high heat. Add the thyme, paprika, celery, bay leaves, carrots, onions, bell peppers, chile peppers and saffron and season with salt and pepper. Cook until the vegetables are soft, 7 to 8 minutes. Add the tomatoes and mash. Add the stock and simmer for 30 minutes. Stir in the chicken. Stir in the picada and olives, then remove from the heat and let stand for 5 minutes.
Ladle the stew into shallow bowls and serve with warm bread for mopping.
Put the chicken in a very large stockpot or divide between 2 pots if necessary. Add the peppercorns, garlic, bay leaves, carrots, celery, lemons, onions and herb bundle and sprinkle with salt. Add enough water to cover the chicken. Bring to a boil, then reduce the heat to low and cook at a rolling simmer for 1 1/2 hours. Remove the chicken from the liquid and let cool. Strain the stock. (You should have 5 to 6 quarts stock.) Remove the skin and bones from the chicken, and shred the meat using your hands or 2 forks.
The stew and the picada can be refrigerated in separate containers in the refrigerator for a make-ahead meal. Reheat the stew over medium heat, adding the picada and olives.