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Boil the potatoes, carrots, Portuguese cabbage (also known as collard greens), and eggs in plenty of water, enough to fully cover everything. Place the codfish on top and bring to a boil.
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Once it starts boiling, let the cod cook for about 10 minutes, until the flesh begins to flake apart. Remove the cod and let the remaining ingredients continue cooking until they’re soft and tender.
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Shred the codfish, removing any bones and skin, and break it into large flakes.
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Meanwhile, grease a large baking dish with olive oil, making sure the bottom is well-coated.
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Drain all the ingredients very well. Set aside 4 of the boiled potatoes to make the mashed potatoes later.
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Slice the remaining potatoes and the boiled eggs into rounds. Cut the cabbage into medium-sized pieces.
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In the baking dish, layer the sliced potatoes first, followed by the eggs and carrots. Then, spread the shredded codfish on top, and finish with the cabbage.
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Drizzle generously with olive oil—about half a liter (2 cups).
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Carefully spread the mashed potatoes you prepared over the top, without mixing the layers underneath.
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Brush the top of the mashed potatoes evenly with the beaten egg.
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Bake in the oven for about 20 minutes, until golden on top.