In the bowl of a food processor fitted with the metal blade, combine the egg, egg yolk, lemon juice, mustard, garlic, salt, and pepper. Process on high speed until smooth, light yellow, and frothy, about 1 minute. While the processor is still running, combine the vegetable and olive oils in a measuring cup with a pour spout, and working very slowly, add the oil to the processor in a thin, steady stream, processing until the oil is completely incorporated and a thick emulsion is formed. (It is very important that the oil is added very slowly, especially at the beginning; otherwise the mayonnaise may break.)

Transfer the mayonnaise to a nonreactive bowl, and add more salt, pepper, and/or lemon juice if desired. Use immediately, or refrigerate in a nonreactive airtight container for up to 1 day.