To start, lightly score each potato along its circumference. Add to a large pot with enough cold, salted water to cover. Bring to a boil then lower heat to a steady simmer until the potatoes are cooked through, about 20 to 30 minutes depending on the size. Pierce with a sharp knife to check for doneness. Drain and set aside.
When cool enough to handle, slip off the peel, which should have loosened during cooking. Pass through a potato ricer or food mill into a large mixing bowl.
Add vegetable oil, lime juice, aji amarillo paste, salt and pepper to taste and mix until thoroughly combined. Chill until ready to use.
Combine oil-packed tuna, lime juice, mayonnaise and chopped onions in a large bowl and mix well. Chop the tomatoes and sprinkle the avocado with additional lime juice. Set aside until ready to use.
Line a baking sheet with plastic wrap. Using an offset spatula, spread the potato mixture in an even layer. Top with tuna salad, tomatoes and avocadoes or other desired filling, leaving a two-inch border along the sides.
Starting with the long end closest to you, lift the causa using the plastic wrap and roll the loaf tightly and evenly, enclosing the filling as you go, jelly-roll style. Wrap the rolled causa in plastic wrap, seam side down. Transfer to serving dish. Refrigerate until firm, about 2 hours.
As an alternative, you can make individual causas by using a cookie cutter (at least 2 inches high) or the outer ring of a springform pan. Add alternating layers of potato and filling (ending with a layer of potato). Gently lift the cutter up and away to serve.
Serve cold as an appetizer or light lunch alongside sliced hard-boiled eggs, avocado dressed with lime juice, pimentos, olives and fresh cheese.