A French Habit
Americans don't usually eat hard-boiled eggs with mayonnaise on them. My partner says it seems wrong, redundant because the mayonnaise is also largely made of eggs. But in France oeufs mayonnaise is a very common appetizer. I think the main difference is that American mayonnaise has no flavor, and the French mayonnaise is delicious. So for this recipe, make your own mayonnaise. If you don't want to eat it on hard-boiled eggs, it will also be good on sandwiches or macaroni salad.
Homemade mayonnaise has raw eggs in it, so don't leave it out for too long. Babies and anyone whose immune system is weak shouldn't eat homemade mayonnaise.
How to make mayonnaise:
(takes about half an hour, but you can do it ahead of when you want it)
Separate one egg, keeping the white to make a meringue later if you like, and put the yolk in a small bowl. With a fork, stir the yolk until it turns light yellow. Mix in a pinch of kosher salt, a teaspoon of regular mustard, and a teaspoon of vinegar. Keep the vinegar bottle handy.
Pour a cup measure full of olive oil (or it's really easier to pour a two cup measure half full of olive oil). Very very slowly, add a few drops of oil at a time to the egg, whisking the egg constantly with a fork. If you start to see puddles of oil, you're going too fast. Let the egg absorb each few drops of oil before you add any more. After about a quarter of a cup, the mayonnaise will get a little thicker and lighter in color, and you can relax a little.
Now add the oil in a constant stream, but a very thin stream, still slowly and still stirring constantly. When you have added half the oil, stop and slowly add another two tablespoons of vinegar and mix that in well.
Now go back to adding oil until all of the oil is mixed in. Taste and add more salt or mustard as needed. You can eat your mayonnaise as a sauce on your hard-boiled eggs right away with lettuce on the side, or let it sit on the counter for one or two hours to thicken, then refrigerate. In any case, you should eat your mayonnaise within a few days.