Gather and measure the ingredients you need. Set up a food processor to incorporate them together.
Using a knife and cutting board, chop the herbs and vegetables (capers, cornichons, chives, tarragon, and parsley). Add these to the work bowl of the food processor along with the egg yolks, vinegar, and dijon mustard.
Turn on the food processor and slowly add the oil in a thin stream to the mixture. (Don't rush the emulsification with the oil. Too much fat, too quickly, and the sauce will break.) Once all the oil is incorporated, remove the sauce from the work bowl into a mixing bowl. Add the reserved diced egg whites and stir to combine. Add the salt and pepper to taste. For a slight spicer, the product adds a dash or two of Tabasco-style hot sauce, optional.
You can serve the sauce immediately or store it in a container with a tight-fitting lid in the refrigerator. The sauce is great on steamed vegetables, fish, chicken, or beef. The sauce will keep fresh for 4-5 days sealed in the refrigerator.