In a medium bowl set on a non-skid mat or folded damp kitchen towel, whisk together the egg yolks, mustard and ¼ teaspoon salt. While whisking constantly, begin by very slowly dribbling in oil until the mixture begins to thicken and emulsify. At this point, while whisking constantly, slowly stream in oil until about one-third of it has been added; the mixture will be very thick. Whisk in 1 tablespoon cold water; the mixture will become pale yellow and looser. Again while whisking, add half of the remaining oil in a thin, fine stream, then whisk in another 1 tablespoon water to loosen. While whisking, stream in the remaining oil, then whisk in the lemon juice. Finally, whisk in 1 to 2 tablespoons cold water, adding enough so the mayonnaise is glossy pale yellow with a pudding-like consistency. Taste and season with salt. Cover and set aside at room temperature while you cook the eggs or refrigerate for up to 5 days.

Fill a large saucepan with 3 inches of water; bring to a boil over medium-high. Using a slotted spoon, lower the whole eggs into the water. Cook, adjusting the heat as needed to maintain a vigorous simmer but not a hard boil, 8½ to 9 minutes for medium yolks or 12½ to 13 minutes for hard-cooked yolks. While the eggs cook, fill a large bowl with about 4 cups ice and 2 cups water; set near the stove.

When the eggs are done, use the slotted spoon to quickly transfer them to the ice bath; let cool for 3 minutes, then remove (if left in the ice bath for longer, the eggs may be more difficult to peel and will be too cold for serving). Gently tap each egg against the counter, turning it so the shell cracks all around. Peel away the shell along with the thin membrane just underneath.

Slice each egg in half lengthwise. If desired, dab a little mayonnaise on the rounded side of each egg half to prevent sliding, then place the halves cut sides up on a platter. Spoon about 1½ teaspoons of the mayonnaise onto each.