Set a stainless steel or glass bowl on top of a pot of simmering water. Ensure the water does not touch the bottom of the bowl.
Whisk the eggs. In the bowl, whisk together the egg yolks and lemon juice until very frothy. Whisk in the Dijon mustard.
Heat the egg mixture over the double boiler, whisking constantly, until the mixture is lightened in color and thickened, about 3 minutes. Do not let the eggs scramble or get too hot.
In another pan or microwave, melt the butter. Slowly add the butter into the egg mixture while whisking constantly. Slowly drizzle in the butter while whisking, stop, and then gradually continue to incorporate all of the butter. Take your time here or the emulsion will separate.
Continue to whisk the sauce until it is velvety, smooth and thickened. Remove from heat and season with salt and cayenne pepper.
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