A good-quality, store-bought mayonnaise can also be used as a base: Whisk in the mustard, lemon juice, and garlic.
In the bowl of a food processor, combine the mustard, egg, lemon juice, and salt; process 5 seconds. While processor is running, pour oil through feed tube in a slow, thin, steady stream; process until ingredients have been amalgamated. Transfer mayonnaise to a medium bowl; stir in garlic. Store in an airtight container, refrigerated, up to 2 days.
Raw eggs should not be used in food prepared for pregnant women, babies, young children, the elderly, or anyone whose health is compromised.