Soak the cod in cold water in the refrigerator for 12 to 24 hours, changing the water several times. Drain, simmer in fresh water for 8 minutes, cool, and flake.
Fry the potato matchsticks in batches until pale golden and crisp. Drain well.
Cook the Brás:
Warm the olive oil in a large skillet. Cook the onion and garlic until soft and translucent, about 6 minutes.
Fold in the flaked cod and cook for 2 minutes. Add the potatoes and toss gently.
Add Eggs and Serve:
Lower the heat and pour in the beaten eggs. Stir gently until just set and still creamy.
Fold in the olives and parsley. Serve immediately with lemon wedges.
Notes
Desalting time depends on the thickness of the cod.