Combine the egg, lemon juice, lime juice, mustard, and onions in a food processor or a blender. Process until smooth, about 15 seconds. With the machine running, slowly add the oil in a steady stream. The mixture will thicken. Add the salt and pepper and pulse for 10 seconds.
Pour the sauce into a decorative bowl, cover, and chill for one hour before serving.
Yield: about 1 1/4 cups