Step 1: Rehydrate the Bacalhau: If using dried salted cod, rinse it thoroughly under cold water to remove surface salt. Place the cod in a large bowl, cover completely with cold water, and refrigerate for 24-48 hours, changing the water every 6-8 hours. The thicker the cod, the longer it will need to soak. Once rehydrated, gently flake the cod, checking for any small bones.
Step 2: Prepare Your Potatoes: If using fresh potatoes, peel and julienne them into thin matchsticks. Rinse them in cold water to remove starch, then pat them completely dry. In a large skillet, heat enough vegetable oil or a mixture of olive and vegetable oil to deep-fry or shallow-fry the potatoes until golden brown and crispy. Remove with a slotted spoon and drain on paper towels. Season lightly with salt. If using pre-packaged potato sticks (batata palha), have them ready.
Step 3: Sauté Aromatics: In a large, deep pan or a large skillet, heat a generous amount of good quality olive oil over medium heat. Add the thinly sliced onions and cook slowly, stirring occasionally, until they are very soft and translucent, about 8-10 minutes. Add the minced garlic and cook for another 1-2 minutes until fragrant, being careful not to burn it.
Step 4: Combine Cod and Potatoes: Add the flaked bacalhau to the pan with the onions and garlic. Sauté for about 5 minutes, gently stirring to combine the flavors. The cod should be heated through and slightly integrated with the onions. Next, gently fold in the crispy fried potatoes. Be careful not to mash them; you want to maintain some of their crispiness.
Step 5: Add the Eggs: In a separate bowl, whisk the eggs with a pinch of salt and pepper. Pour the whisked eggs over the cod and potato mixture in the pan. Reduce the heat to low-medium and gently stir with a wooden spoon or spatula. Cook just until the eggs are set but still creamy and moist. The key is not to overcook the eggs; they should be soft and custardy, not dry.
Step 6: Finish and Serve: Remove the pan from the heat immediately. Stir in the freshly chopped parsley. Transfer the Bacalhau à Brás to a serving platter. Garnish generously with additional fresh parsley and black olives. Serve hot.