With processor running, drop garlic through food chute. Process until minced; add egg substitute and next 5 ingredients. With processor running, slowly pour oils in a fine stream through food chute. Process until thickened, stopping once to scrape down sides. Add basil; process until minced. Cover and chill.
Note: Basil Aïoli makes a tasty spread. Try it instead of mayonnaise on your next tomato sandwich.