Begin with all ingredients at room temperature, they will emulsify more readily.
In a food processor or blender, process the yolks, salt, mustard, cayenne, lemon juice and water together to form a stable base. With the motor running, add a few drops of the oil, the mixture will begin to thicken.
Then, begin adding the remaining oil in a thin, steady stream. Scrape down the sides of the processor/blender. Check seasoning. Keep covered in the refrigerator for up to 3 days.
*RAW EGG WARNING
Food Network Kitchens suggest caution in consuming raw and lightly-cooked eggs due to the slight risk of Salmonella or other food-borne illness. To reduce this risk, we recommend you use only fresh, properly-refrigerated, clean, grade A or AA eggs with intact shells, and avoid contact between the yolks or whites and the shell.