Mince and mash the garlic to a paste with the salt. In a food processor or blender blend together the basil, the egg, the lemon juice and the garlic paste, with the motor running add the oils in a slow stream, and blend the mayonnaise well. Season the mayonnaise with salt and pepper.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight 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.
Recipe courtesy of Gourmet Magazine