Combine the Base: In a medium bowl (for hand-whisking) or the container of your blender or food processor, combine the egg yolk and Dijon mustard.
Whisk or pulse briefly to combine.
Emulsify the Oil: This is the crucial step! Begin adding the oil extremely slowly—literally drop by drop at first—while whisking vigorously or with the machine running.
Once you see a thick emulsion begin to form, you can start adding the oil in a thin, steady stream.
Continue until all the oil is incorporated and the mayonnaise is thick and glossy.
Season and Finish: Whisk in the lemon juice (or vinegar), salt, and pepper.
Taste your creation and adjust the seasoning as needed. Store in an airtight container in the refrigerator for up to one week.