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1
Submerge the salt cod in a large vessel of cold water and refrigerate for 36–48 hours, changing the water every 8 hours. At the end of this period, tear a small piece from the thickest part and taste it: the flesh must be pleasantly seasoned throughout, with no aggressive salinity remaining.
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2
Drain the desalinated cod and place it in a wide pan. Cover with fresh unsalted water and bring slowly to 70°C — never higher, never a boil. Hold at that bare tremor for 10 minutes. The fish is ready when the flesh yields to gentle pressure and separates into large, moist flakes; it must not be firm or dry at the centre.
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3
Lift the cod from the poaching water, remove all skin and bone, and flake the flesh while still warm. If making the Provençal version, rub the interior of a heavy pan with the crushed garlic clove before the fish goes in. Place the flaked cod in the pan over the lowest possible heat.
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4
This is where the dish lives or dies. With the pan held over very low heat to keep the contents warm throughout, begin adding the olive oil — warmed to 40°C — in the thinnest possible stream, working the mixture vigorously and constantly with a wooden spatula exactly as one mounts a mayonnaise. Alternate small additions of warm milk with the oil, fully incorporating each addition before the next. If at any point the emulsion looks heavy or greasy, add a splash of warm milk and beat vigorously to restore it. Cold ingredients or a moment's inattention will break the emulsion.
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5
Continue working until the brandade is snow-white and impossibly smooth, and holds soft peaks when the spatula is lifted. It must be light and cloud-like — not dense, not pasty, not stiff.
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6
To serve in the classic manner, mound the brandade on a warmed platter and surround with croûtons fried in olive oil. For the gratiné presentation, spoon into individual ramekins, dust with the combined breadcrumb-and-Parmesan crust, and pass under a hot salamander until the surface is golden and bubbling at the edges.