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1
Set a small, light-coloured saucepan over medium heat. Add the unsalted butter. It will melt, then produce a vigorous foam as the water in the butter evaporates. Do not stir. The foam is a marker: while it persists, the milk solids are not yet colouring.
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2
As the foam subsides, watch the base of the pan closely. The milk solids will progress from white to pale gold. For beurre noisette, remove the pan from the heat the instant the solids reach the colour of a blanched hazelnut — a deep, even gold. Simultaneously, the butter will smell of hazelnuts, caramel, and a faint toasty note. This is where the dish lives or dies: at the correct colour, the butter is ten seconds from correct to burnt in a hot pan; residual heat will continue the Maillard reaction for thirty seconds after the pan leaves the flame without the acid to stop it.
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3
For beurre noisette: off the heat, add a squeeze of fresh lemon juice immediately. The acid arrests the browning by pH change. Pour at once over the dish, or pass through a fine sieve. Serve without delay.
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4
For beurre noir: allow the solids to pass one stage beyond the noisette colour, to a dark brown. The butter must not be allowed to turn black — black denotes burnt, and a burnt butter is bitter and unusable. Off the heat, add the red wine vinegar or sherry vinegar; it will produce a violent, fragrant sizzle. Add the drained capers. Pour immediately over the prepared dish — skate, brain preparations, or poached eggs in the classical tradition.