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For the crisps, peel parsnips and slice lengthways, using a mandolin to slice as thinly as possible. Deep-fry in hot oil until crisp. Drain on kitchen towels to absorb any excess fat and set aside in a dry place. Can be made up to 1 day ahead and kept in an airtight container.
For the caramelised garlic, peel the shallots and garlic. Blanch the shallots in boiling salted water for 10 mins or until tender. Cut them in half if large and drain well. Put the garlic in a small saucepan of boiling salted water, bring to the boil for 2 mins, then drain and change the water. Repeat 4 times and drain well. Heat a little olive oil in a frying pan over a medium heat, add the shallots and garlic and cook until caramelised, shaking the pan so they don’t stick. Can be made in the morning and reheated.
For the purée, peel the parsnips and cut them into big chunks. Cook in boiling salted water until tender. Bring the milk to the boil and set aside. Drain the parsnips well, then put in a blender with the butter and some of the boiled milk and blend until totally smooth: the purée should be the consistency of double cream, so add more milk if necessary. Season and keep warm or make ahead and reheat.
Scale and fillet the fish. Remove any pin bones (or ask your fishmonger to do this, but keep the bones for the sauce). Rinse under cold water and pat dry with kitchen paper. Score the skin of the fish several times with a sharp knife to help prevent it from curling during cooking. Leave the bones (but not the heads) to soak in cold water.
To make the sauce, heat a little olive oil in a pan and cook the shallots for about 5 mins until golden and soft. Add the mushrooms and continue to cook for 10 mins, stirring occasionally. Drain the fish bones, add to the pan and cook for 5-6 mins. Add the vinegar and let it come to the boil for 3 mins, then add the stock, season lightly and simmer for 30 mins, skimming at regular intervals. Pass through a fine sieve into a clean pan, bring back to the boil and whisk in the butter to thicken and gloss the sauce.
Heat a non-stick frying pan until smoking hot, add a few drops of oil, then add the fish, skin side down. Season with salt and pepper and press the fish down with a palette knife if it begins to curl up. Once the skin is well browned, turn the fillets over and cook the other side; the whole process should take no more than 5-6 mins, depending on the thickness of the fish. To serve, spoon the purée on to warmed plates, make a hollow in the centre, and fill with the caramelised garlic and shallots. Pour the sauce around the purée, place the fish on top and add a few parsnip crisps for garnish.
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