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1
Lay the fillets skin-side up on a rack and press kitchen paper firmly against every surface, inside and out, so that any surface moisture is drawn away before the pan sees the fish — wet skin steams rather than crisps. Allow to air for at least 10 minutes. Score the skin at 3–4 points with a sharp knife — shallow cuts perpendicular to the length of the fillet — to prevent curling under heat, then season the flesh side lightly with fine sea salt immediately before cooking.
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2
Heat a large heavy frying pan over high heat until the surface is very hot, then add the clarified butter, which should shimmer and begin to move immediately — clarified butter carries no milk solids to burn, so it holds the heat needed to set the skin before the flesh overcooks. Lay the fillets skin-side down and, in the first 30 seconds, apply gentle, even pressure across each fillet with a fish spatula to maximise skin contact with the pan and prevent curling. Do not move the fillets; cook skin-side down for 4 minutes, and know the skin is ready when it releases from the pan of its own accord and is uniformly golden and rigid to the touch at the edges — forced movement before this point will tear it.
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3
Flip each fillet and cook flesh-side down for 90 seconds only, because salmon trout carries so little thickness on the flesh side that another minute past this point turns the interior chalky and dry. The flesh should feel just-set and yielding under light finger pressure — not firm, which indicates overcooking. Transfer immediately to warm plates and allow to rest while the sauce is made.
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4
In a small clean pan over medium-high heat, melt the whole butter and continue cooking, swirling frequently, until the milk solids turn deep golden and the butter smells of hazelnuts — the swirling keeps the solids in suspension so they colour evenly rather than sinking and scorching on one patch of the pan. Add the sage leaves; they will crisp within 20–30 seconds. The butter should be amber, not dark brown, and carry a clean, nutty smell with no acrid note.
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5
Remove the pan from heat, then grate a little lemon zest directly over the resting fillets so the oils fall onto the fish rather than being lost to a board. Pour the brown butter and sage leaves immediately over the fish. Serve at once on warm plates — the dish loses heat within 2 minutes.