Place the egg, egg yolk, garlic and lemon juice in the food processor. (You can make it by hand, but it is much easier and more fail-safe in the food processor.) While it is blending, slowly drizzle in the oil, until it is thick and creamy. Add the cayenne pepper, sea salt and pepper.
Did you add the oil too fast? If you add the oil too fast, the aioli will separate, so go very slowly. If it does separate, take the mixture out of the food processor, saving the separated aioli, and start over. Put another egg and egg yolk into the food processor and blend them. Slowly, slowly add the separated aioli. That should do the trick.
This more than what you need for these fries but great for sandwiches and such, too
**If you don't want to eat raw eggs - skip the eggs and the oil and 1 cup of your favorite mayo instead. Yours will not have this bright yellow color, which comes from the yolks on our beautiful farm fresh eggs. If you are vegan, substitute your favorite vegan mayo instead.