1. Hard-boil the eggs for 10 minutes, cool in cold water, peel and chop finely.
2. In a wide pan, bring the water to the boil with the sliced carrot, parsley root, halved onion, bay leaves, allspice and salt and simmer for 15 minutes to make a flavourful vegetable stock.
3. Lower the heat so the liquid is barely trembling, slide in the fish cut into portions and poach without boiling for 8-10 minutes, until the flesh is white and flakes easily.
4. Meanwhile melt the butter in a small frying pan and heat until pale hazelnut-brown and fragrant; sprinkle in the breadcrumbs and toast for half a minute.
5. Off the heat, stir the chopped egg and finely chopped dill into the butter and season with a little lemon juice.
6. Carefully lift the fish onto plates with a slotted spoon, spoon plenty of the hot egg and dill butter over it and serve with boiled new potatoes and lemon wedges.
Tip: "Po polsku" (Polish style) is the characteristic trio of browned butter, breadcrumbs and hard-boiled egg – cauliflower and asparagus are served the same way.
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