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1
Season the fish on both sides with fine salt and white pepper. At the absolute last second before placing in the pan, dredge each piece in plain flour and shake off all excess firmly — only the thinnest, even film should remain. Excess flour absorbs butter, turns pasty, and burns in patches.
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2
Heat the clarified butter in a wide, heavy pan over moderate-high heat to 170–180°C. Lay the fish presentation-side down in the foaming butter without crowding. Do not move it. Cook for 3–4 minutes; the dish lives or dies here — the crust must set undisturbed until it releases cleanly from the pan and shows an even, deep golden colour beneath.
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3
Turn once with a palette knife and cook a further 2–3 minutes. The flesh should be just opaque through to the bone and yield slightly when pressed at its thickest point. Transfer to a warm oval platter.
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4
Wipe the pan clean. Add the whole butter and return to the heat. Cook, swirling, until the butter foams, the foam subsides, and the milk solids turn hazelnut brown — beurre noisette, at approximately 150°C. The colour must not pass hazelnut: beyond that point lies beurre noir, which is acrid and unfit for use.
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5
Remove the pan from the heat and immediately add the lemon juice — it arrests the browning and creates a light emulsion with the butter — then the chopped flat-leaf parsley. Sprinkle fleur de sel over the fish. Pour the sizzling beurre noisette over the fish at the table: the audible hiss on contact is part of the dish. Serve instantly.