To make the sauce, sweat the mixture of carrot, leek, celery and onion in a large pan with half the butter, until soft. Add the cognac and let it boil, then add all the rest of the sauce ingredients, except the remaining butter and basil leaves. Simmer for 30 minutes then pass through a fine sieve. Bring the sauce back to the boil and simmer until it has reduced to about 150ml/5fl oz.
Bring the butter beans to the boil in a large pan of salted water. Simmer gently until very soft. Remove from the heat and keep warm in the cooking liquid.
Place the eggs in boiling water and boil for 8 minutes. Drain, remove the shells and keep warm.
Remove the outer leaves of the fennel but don't cut off the tops. Slice into thin sections then cook in salted water until just tender. Drain and keep warm.
Preheat the oven to 230C/450F/Gas 8. Roast the cod in the oven until just cooked through. This will take about 10-15 minutes depending on the thickness of the fillets. Place the cod on four warmed plates. Drain the butter beans and divide between the plates. Add the fennel, then cut the eggs in half and put one half on each plate, then add a spoonful of aioli to each serving.
Bring the sauce to the boil and whisk in the last 25g/1oz of butter, then add the basil leaves. Pour the sauce over the butter beans, egg and fennel and serve.
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