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1
Melt the butter for the velouté in a heavy saucepan over medium heat. Add the flour and cook, stirring constantly, for 3 minutes — the roux should be pale blond and smell of cooked pastry, with no raw-flour scent remaining.
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2
Add the cold white chicken stock gradually, whisking after each addition to prevent lumps. Bring to a gentle simmer and cook for 30–40 minutes, skimming the surface at regular intervals, until the velouté tastes deeply of chicken with no trace of flour on the palate.
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3
While the velouté simmers, sweat the sliced mushrooms in butter in a small pan over medium heat without allowing them to colour, until they release their liquid fully. Strain the liquid through a fine sieve, reserving only the clear mushroom essence; discard the solids.
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4
Combine the finished velouté, the mushroom essence, and the heavy cream in a wide saucepan. Reduce over medium heat, stirring frequently, until the sauce coats the back of a spoon and a clean line drawn through it with a finger holds its edges — a reduction of roughly one-third. This is where the dish lives or dies: insufficient reduction leaves the sauce thin and cream-heavy; excessive reduction produces a sauce so thick it clumps on the plate.
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5
Remove the pan from the heat entirely. Add the cold butter piece by piece, swirling the pan continuously rather than whisking, until each piece has emulsified into the sauce and the surface is glossy.
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6
Pass the sauce through a fine chinois. The finished Suprême should be ivory-white, glossy as satin, and flow from a spoon in a thick, unbroken ribbon.