Combine the egg, lemon juice, basil, green onions, salt, and pepper in a food processor or blender and purée for 15 seconds. While the processor is running, slowly steam in the oil.
When all of the oil has been added, turn off the machine and scrape down the sides and cover. Process again until the mixture becomes a thick mayonnaise. Remove the mayonnaise to an airtight container and refrigerate for at least 30 minutes. Use within 24 hours.
Yield: 1 cup