Place the two egg yolks, the vinegar (or lemon juice), mustard, salt and white pepper in the food processor bowl. Pulse several times to combine the ingredients. Turn the processor on and through the feed tube, pour about 1/3 of the oil in, in a very slow stream. The eggs and oil will begin to emulsify and thicken. Stop the processor and remove the lid. With a rubber spatula scrape down the sides, and if there is any raw egg near the center bottom of the bowl, scrape it outward toward the rest of the sauce.
Put the lid back on and turn the machine back on. Through the feed tube, pour the rest of the oil in, in a very slow stream.
Once you have added the oil and the sauce is thick, taste for seasoning. Add salt and pepper or even an extra bit of vinegar or lemon juice to your personal taste.
If the mayonnaise gets too thick, whisk in a tablespoon of ice water, one at a time, until it reaches the desired consistency.