For the second variation: Soften 2 chopped onions in 2 tablespoons of olive oil in a heavy, covered pan. Cut 900g red-skinned potatoes into quarters and boil in 1.75 liters of water until just soft. Drain, reserving the water, and cut potatoes into small pieces. Add another tablespoon of oil to the onions, then add potatoes. Stir in stoned black olives and anchovy fillets. Add reserved potato water, bring to a boil, and simmer for 5-10 minutes before liquidizing.
Storage: Store leftovers in an airtight container in the refrigerator for up to 3 days. Reheat gently on the stovetop, adding a splash of water if needed to adjust consistency.
Substitutions: You can omit anchovies for a strictly vegetarian version, though you may lose some umami depth. Consider adding a dash of soy sauce or miso paste instead.