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1
Sweat the finely chopped shallots in 15 g butter over low heat until soft and translucent but without colour. Add the white wine and tarragon vinegar. Raise the heat and reduce by two-thirds; the liquid will be syrupy and the sharp vinegar edge will have mellowed to a concentrated perfume.
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2
Add the warm fish velouté to the reduction. Simmer gently for 10 minutes, skimming as needed. Pass through a fine chinois, pressing the shallots lightly. The strained sauce should be smooth, ivory-coloured, and coat the back of a spoon.
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3
Prepare the beurre vert. Drop the tarragon and chervil leaves into rapidly boiling water for 10 seconds precisely, then transfer immediately to ice water. Drain, spin dry in a salad spinner until no surface moisture remains. Pound in a mortar with the softened butter — and the spinach juice if using — until the mixture is absolutely smooth and vividly, evenly green. This step decides the dish: under-blanched herbs yield dull colour and raw bitterness; over-wet herbs cloud the butter; too little pounding leaves streaks. The finished butter must be the colour of new sage leaves and uniformly smooth.
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4
Remove the strained sauce from the heat entirely. Whisk the beurre vert into the sauce in small additions until the colour shifts from ivory to a subtle, even jade-green. The sauce must not return to the boil; heat will brown the colour and turn the herb flavour bitter. Pass once more through a fine chinois for absolute smoothness. The finished sauce should be pale jade — noticeably green, never vivid, and wholly unlike pesto.