Add the chicken, water or stock, halved onion, crushed garlic, bay leaf, salt, and peppercorns to a saucepan. Bring to a gentle simmer over medium heat. Once it reaches a simmer, reduce the heat to low and poach gently for 14–18 minutes, depending on thickness, until the chicken is just cooked through and reaches 74°C internally. Do not boil aggressively or the chicken will tighten and dry out. Remove the chicken to a plate and let it cool just enough to handle. Strain and reserve 500ml of the cooking liquid. Shred the chicken into medium fine strands by hand or with two forks. The pieces should be long and soft rather than chopped into cubes. This gentle poach protects two things at once — the chicken's own tender texture, and a clean, flavourful cooking liquid that gets used repeatedly throughout the rest of the recipe to build and adjust the sauce.

While the chicken cooks, peel the potatoes and cut them into thick rounds, about 1.5cm thick. Place them in a saucepan, cover with cold salted water, and bring to a boil. Reduce to a steady simmer and cook for 12–15 minutes until just tender. A knife should slide in cleanly, but the slices should still hold their shape without breaking. Drain carefully and set aside. Rinse the rice until the water runs mostly clear. Add the rice, water, oil or butter, and salt to a saucepan. Bring to a boil, stir once, cover, reduce to very low, and cook for 15 minutes. Turn off the heat and let it rest covered for 10 minutes, then fluff with a fork. The rice should be tender and separate, suitable for molding or spooning alongside the sauce. Using genuinely waxy potatoes here, rather than a starchier variety, is what lets the slices hold their shape through both boiling and later being topped with hot sauce — a floury potato would risk breaking apart under that same handling.

Tear the bread into pieces and place it in a blender. Add the evaporated milk, walnuts or pecans, and 120ml of the reserved chicken broth. Let it sit for 2–3 minutes so the bread softens fully, then blend until completely smooth. The mixture should be creamy and pourable, with no visible bread lumps or nut grit. If it seems too thick to blend smoothly, add a small splash more broth. That brief resting period before blending gives the bread real time to soften and absorb the surrounding liquid — blending it while still dry and firm would leave the blender working against tougher, less cooperative pieces, risking a less smooth final result.

Heat the oil in a wide sauté pan or deep skillet over medium heat. Add the finely diced onion and cook for 6–8 minutes, stirring often, until softened and lightly golden at the edges. Add the minced garlic and cook for 30–45 seconds until fragrant. Add the ají amarillo paste and cook for 3–4 minutes, stirring constantly. This step is important: the paste should darken slightly, smell sweeter, and lose its raw sharpness. The oil may take on a rich golden-orange color. This active cooking stage is what transforms the ají paste from something sharp and raw-tasting into a genuinely rounded, sweeter flavour base — that transformation is what gives the finished sauce both its real depth and its vivid golden colour.

Pour the blended bread mixture into the pan with the ají base. Stir immediately to combine. Add about 180ml more reserved chicken broth and bring the sauce to a very gentle simmer. Stir often as it thickens. Add the grated cheese, black pepper, and salt. Simmer gently for 5–8 minutes, stirring regularly, until the sauce becomes thick, creamy, and glossy. It should coat the spoon heavily and slowly settle back into the pan when stirred. If it becomes too thick or tight, loosen it with a little more reserved broth. If it is too loose, continue simmering gently until it thickens. The finished sauce should be vivid golden-yellow, rich but not gluey, with enough body to cling to the chicken and sit neatly over the potatoes. Using the chicken's own reserved cooking liquid to build and adjust this sauce, rather than plain water, means genuine chicken flavour is worked directly into the base at multiple points, rather than the sauce relying entirely on the bread, milk, and cheese for its own character.

Add the shredded chicken to the sauce and fold gently to coat every strand. Simmer over low heat for 3–5 minutes until the chicken is fully heated through and the sauce clings to the shreds. Taste and adjust seasoning with extra salt or pepper if needed. The final texture should be creamy and luxurious, with visible shreds of chicken suspended in the sauce rather than drowning in liquid. It should not be runny, and it should not be so thick that it turns pasty. Folding the chicken in gently, rather than stirring vigorously, keeps the shredded strands intact and visibly distinct within the sauce, preserving the texture contrast this dish depends on.

Arrange several slices of boiled potato on each plate as the base. Spoon the ají de gallina generously over the potatoes. Add a portion of white rice alongside. Garnish each serving with half a hard-boiled egg and 2 black olives. Serve hot. Traditionally, the contrast is important: creamy spicy sauce over potatoes, plain rice at the side, and the salty bite of olives and egg as garnish. Keeping the rice genuinely plain and unseasoned, rather than flavoured, is a deliberate choice — its job is specifically to provide a neutral counterpoint to the rich, spiced sauce, not to compete with it.