Place the salt cod in a large dish and pour over the measurement water. Cover and leave to soak in the refrigerator for 24–48 hours, depending on its thickness, changing the water at least twice. Drain, then place the fish in a saucepan and cover with fresh cold water. Heat just to boiling point, then remove the pan from the heat and leave to stand for 15 minutes.
Meanwhile, cook the potatoes in a separate saucepan of boiling salted water for 15 minutes until very tender, then drain.
Drain the salt cold, discard any skin and bones, then flake into small pieces. Place in a food processor with the garlic and half the oil and pulse until it forms a rough purée. Add the potatoes and remaining oil, then pulse again until the mixture forms a fluffy purée. Season with a little salt, if needed.
Spoon the dip into a serving dish and serve with slices of toast.