Whisk together egg yolk, lemon juice, mustard, salt, and 1 tsp cold water in a medium bowl until frothy.
Whisking constantly, add the oil drop by drop at first. Once the emulsion begins to thicken, add the oil in a thin, steady stream, whisking continuously until thick and all oil is incorporated.
Taste and adjust salt or lemon juice. Refrigerate and use within 3–4 days.