Place potatoes in a large pot, cover with salted water, and bring to a boil. Cook until tender, about 20 minutes. Drain and let cool slightly, then peel if not done already.
While potatoes are still warm, mash them in a bowl until smooth. Add oil, lime juice, salt, aji amarillo paste, and black pepper. Mix well and set aside to cool completely.
In a separate bowl, combine shredded chicken, mayonnaise, red onion, and cilantro. Season with salt and pepper to taste.
Line a loaf pan or small bowl with plastic wrap. Spread half of the potato mixture evenly in the bottom, pressing firmly.
Layer the chicken mixture over the potatoes, then top with avocado slices and hard-boiled egg slices if using.
Cover with the remaining potato mixture, pressing gently to seal. Refrigerate for at least 30 minutes to set.
To serve, invert onto a platter lined with lettuce leaves. Garnish with olives and parsley if desired. Slice and serve cold or at room temperature.
Tips
For a vegetarian version, replace chicken with sautéed mushrooms or hearts of palm.
Causa can be made a day ahead and refrigerated; cover tightly with plastic wrap.
If you can't find aji amarillo paste, substitute with a mix of turmeric and cayenne for color and heat.