For a Smoked Aioli, add 3/4 teaspoons hickory smoke powder (sold with spices and other seasonings at the supermarket or online) to the egg mixture before adding the olive oil.
Fill a small saucepan with 3 inches of water and bring to a boil over medium heat. With a slotted spoon, lower the egg into the water and boil for 6 minutes. Remove the egg and put it in a small bowl of cold water until cool enough to handle. Peel the egg.
Put the egg, garlic, lemon juice, Dijon, vinegar, salt, and white pepper into a blender and puree on low speed. With the blender running, add the oil in a steady, slow stream, blending until emulsified and thickened. Add the hot sauce, if using, and blend on high speed for about 1 minute. Transfer to an airtight container and store in the refrigerator for up to 3 days.