Set up. Place a damp towel under a medium bowl so it does not slide. Measure the lemon juice and vinegar into a small cup so you can add them quickly later.
Mix the dry ingredients. Whisk the mustard, salt, powdered sugar, and cayenne together in the bowl.
Add the yolks. Whisk in the yolks until smooth and slightly thickened, then whisk in ½ teaspoon of the vinegar.
Start the oil, drop by drop. Whisk constantly while adding the oil one drop at a time. Spend the first 4 to 5 minutes on the first ¼ cup. The mixture should turn thick, pale, and glossy.
Thin and continue. When it gets very thick, whisk in a little lemon juice or vinegar. Now you can add the oil in a thin, steady stream, alternating with the remaining acid, until all of the oil and acid are used.
Taste and adjust. Add salt or a few drops more lemon juice if needed.
Store. Transfer to a clean jar, refrigerate right away, and use within 4 days.