This is from Alton Brown of FoodTV's Good Eats. I really love this recipe.
This recipe uses raw eggs. If you like you can probably find some kind of pasteurized egg.
2 tblspns White Wine Vinegar (don't overdo)
2 tblspns Lime Juice (fresh squeezed. Don't overdo)
1 egg yolk
1 whole egg (fyi, it's usually best to let eggs come to room temperature)
1 tspn table salt
1 tspn dry mustard (the powdered stuff in the spice aisle)
1/4 tspn sugar
2 to 3 tablespoons chile oil (the stuff I use is just called “hot oil”, brand name of “China Bowl Select”. It comes in a small bottle with an orange label, but you can’t get it at every grocery store. Some grocery stores sells something with a blue label, don’t use it. It’s tasteless.)
Safflower or corn oil added to the chile oil to make 2 cups total oil (put the chile oil into a measuring cup and add safflower or corn oil until it equals 2 cups total).
Add all wet ingredients (except the oil which is a liquid but isn't "wet") to the work bowl of a food processor along with the salt, mustard and sugar. (You must use a food processor. I tried it once in my blender, didn't work at all). Pulse 5 times. Turn processor on and slowly add oil in a steady stream until incorporated. Keep at room temperature for 2 hours. Refrigerate for up to 1 week.
The strange instruction about keeping at room temperature for 2 hours is because Alton disputes the notion that mayo encourages bacteria growth. In fact, Alton says mayo actually hinders bacteria because of the acid. However, it is a better inhibitor of bacteria when it is at room temperature, hence keeping the mayo at room temperature to reduce the likelihood of getting bacteria from the raw eggs. Me, I never bother doing this step.