Drain the yogurt in a fine sieve set over a bowl for about 4 hours.
Put eggs into a small pot with enough water to cover. Cover with lid and bring water to a boil. Cook for exactly 9 minutes (hard-boil). Run eggs under cold water to stop the cooking.
Peel the eggs and cut in half. Scoop out the egg yolks with a spoon and place into a bowl. (Reserve the whites for another use, such as chopped and sprinkled into a salad.)
Add 1-2 tablespoons of water to the egg yolks and mash with a wooden spoon until the yolks are creamy. Mix in cayenne.
Mix in the drained yogurt and season with salt.
Squeeze 1-2 teaspoons of lemon juice into the mayonnaise. Mix in vinegar and refrigerate, covered, 2-3 hours before using.