Rinse the salted cod under cold running water, then pat dry. Check for any remaining bones and remove them.

In a pan over medium heat, warm the olive oil, then add the chopped onion and garlic. Sauté until the onion is translucent and aromatic, about 5 minutes.

Add the cod to the pan, breaking it into smaller pieces. Cook until the cod turns opaque and flakes easily, approximately 10 minutes.

In a mixing bowl, combine the cooked cod mixture with mashed potato, chopped parsley, and the beaten egg. Season with salt and pepper to taste.

Form the mixture into four equal-sized patties, ensuring they hold together well.

In a large skillet over medium heat, fry the patties for about 4 minutes on each side or until golden brown.

Toast the burger buns lightly in the same skillet until golden.

To assemble, spread a tablespoon of aioli on the base of each toasted bun, followed by a lettuce leaf, a slice of tomato, and the cod patty. Top with the other half of the bun.