Place the egg yolks, lemon juice, garlic, salt and pepper in the bowl of a food processor fitted with the steel blade and process until smooth. With the processor running, slowly pour the oils into the bowl through the feed tube and process until emulsified. Add the basil and continue processing until finely pureed. Store the mayonnaise in the refrigerator.
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell. For recipes that call for eggs that are raw or undercooked when the dish is served use shell eggs that have been treated to destroy Salmonella, by pasteurization or another approved method.